A literature review is a critical analysis and synthesis of existing research on a particular topic. It provides an overview of the current state of knowledge, identifies gaps, and highlights key findings in the literature. 1 The purpose of a literature review is to situate your own research within the context of existing scholarship, demonstrating your understanding of the topic and showing how your work contributes to the ongoing conversation in the field. Learning how to write a literature review is a critical tool for successful research. Your ability to summarize and synthesize prior research pertaining to a certain topic demonstrates your grasp on the topic of study, and assists in the learning process.
A well-conducted literature review demonstrates the researcher’s familiarity with the existing literature, establishes the context for their own research, and contributes to scholarly conversations on the topic. One of the purposes of a literature review is also to help researchers avoid duplicating previous work and ensure that their research is informed by and builds upon the existing body of knowledge.
A literature review serves several important purposes within academic and research contexts. Here are some key objectives and functions of a literature review: 2
1. Contextualizing the Research Problem: The literature review provides a background and context for the research problem under investigation. It helps to situate the study within the existing body of knowledge.
2. Identifying Gaps in Knowledge: By identifying gaps, contradictions, or areas requiring further research, the researcher can shape the research question and justify the significance of the study. This is crucial for ensuring that the new research contributes something novel to the field.
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3. Understanding Theoretical and Conceptual Frameworks: Literature reviews help researchers gain an understanding of the theoretical and conceptual frameworks used in previous studies. This aids in the development of a theoretical framework for the current research.
4. Providing Methodological Insights: Another purpose of literature reviews is that it allows researchers to learn about the methodologies employed in previous studies. This can help in choosing appropriate research methods for the current study and avoiding pitfalls that others may have encountered.
5. Establishing Credibility: A well-conducted literature review demonstrates the researcher’s familiarity with existing scholarship, establishing their credibility and expertise in the field. It also helps in building a solid foundation for the new research.
6. Informing Hypotheses or Research Questions: The literature review guides the formulation of hypotheses or research questions by highlighting relevant findings and areas of uncertainty in existing literature.
Let’s delve deeper with a literature review example: Let’s say your literature review is about the impact of climate change on biodiversity. You might format your literature review into sections such as the effects of climate change on habitat loss and species extinction, phenological changes, and marine biodiversity. Each section would then summarize and analyze relevant studies in those areas, highlighting key findings and identifying gaps in the research. The review would conclude by emphasizing the need for further research on specific aspects of the relationship between climate change and biodiversity. The following literature review template provides a glimpse into the recommended literature review structure and content, demonstrating how research findings are organized around specific themes within a broader topic.
Climate change is a global phenomenon with far-reaching consequences, including significant impacts on biodiversity. This literature review synthesizes key findings from various studies:
Climate change-induced alterations in temperature and precipitation patterns contribute to habitat loss, affecting numerous species (Thomas et al., 2004). The review discusses how these changes increase the risk of extinction, particularly for species with specific habitat requirements.
Observations of range shifts and changes in the timing of biological events (phenology) are documented in response to changing climatic conditions (Parmesan & Yohe, 2003). These shifts affect ecosystems and may lead to mismatches between species and their resources.
The review explores the impact of climate change on marine biodiversity, emphasizing ocean acidification’s threat to coral reefs (Hoegh-Guldberg et al., 2007). Changes in pH levels negatively affect coral calcification, disrupting the delicate balance of marine ecosystems.
Recognizing the urgency of the situation, the literature review discusses various adaptive strategies adopted by species and conservation efforts aimed at mitigating the impacts of climate change on biodiversity (Hannah et al., 2007). It emphasizes the importance of interdisciplinary approaches for effective conservation planning.
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Writing a literature review involves summarizing and synthesizing existing research on a particular topic. A good literature review format should include the following elements.
Introduction: The introduction sets the stage for your literature review, providing context and introducing the main focus of your review.
Body: The body of the literature review is where you provide a comprehensive analysis of existing literature, grouping studies based on themes, methodologies, or other relevant criteria.
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Conclusion: The conclusion of your literature review should summarize the main findings, highlight the contributions of the review, and suggest avenues for future research.
Conducting a literature review is an essential step in research that involves reviewing and analyzing existing literature on a specific topic. It’s important to know how to do a literature review effectively, so here are the steps to follow: 1
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The literature review sample and detailed advice on writing and conducting a review will help you produce a well-structured report. But remember that a good literature review is an ongoing process, and it may be necessary to revisit and update it as your research progresses. By combining effortless research with an easy citation process, Paperpal Research streamlines the literature review process and empowers you to write faster and with more confidence. Try Paperpal Research now and see for yourself.
A literature review is a critical and comprehensive analysis of existing literature (published and unpublished works) on a specific topic or research question and provides a synthesis of the current state of knowledge in a particular field. A well-conducted literature review is crucial for researchers to build upon existing knowledge, avoid duplication of efforts, and contribute to the advancement of their field. It also helps researchers situate their work within a broader context and facilitates the development of a sound theoretical and conceptual framework for their studies.
Literature review is a crucial component of research writing, providing a solid background for a research paper’s investigation. The aim is to keep professionals up to date by providing an understanding of ongoing developments within a specific field, including research methods, and experimental techniques used in that field, and present that knowledge in the form of a written report. Also, the depth and breadth of the literature review emphasizes the credibility of the scholar in his or her field.
Before writing a literature review, it’s essential to undertake several preparatory steps to ensure that your review is well-researched, organized, and focused. This includes choosing a topic of general interest to you and doing exploratory research on that topic, writing an annotated bibliography, and noting major points, especially those that relate to the position you have taken on the topic.
Literature reviews and academic research papers are essential components of scholarly work but serve different purposes within the academic realm. 3 A literature review aims to provide a foundation for understanding the current state of research on a particular topic, identify gaps or controversies, and lay the groundwork for future research. Therefore, it draws heavily from existing academic sources, including books, journal articles, and other scholarly publications. In contrast, an academic research paper aims to present new knowledge, contribute to the academic discourse, and advance the understanding of a specific research question. Therefore, it involves a mix of existing literature (in the introduction and literature review sections) and original data or findings obtained through research methods.
Literature reviews are essential components of academic and research papers, and various strategies can be employed to conduct them effectively. If you want to know how to write a literature review for a research paper, here are four common approaches that are often used by researchers. Chronological Review: This strategy involves organizing the literature based on the chronological order of publication. It helps to trace the development of a topic over time, showing how ideas, theories, and research have evolved. Thematic Review: Thematic reviews focus on identifying and analyzing themes or topics that cut across different studies. Instead of organizing the literature chronologically, it is grouped by key themes or concepts, allowing for a comprehensive exploration of various aspects of the topic. Methodological Review: This strategy involves organizing the literature based on the research methods employed in different studies. It helps to highlight the strengths and weaknesses of various methodologies and allows the reader to evaluate the reliability and validity of the research findings. Theoretical Review: A theoretical review examines the literature based on the theoretical frameworks used in different studies. This approach helps to identify the key theories that have been applied to the topic and assess their contributions to the understanding of the subject. It’s important to note that these strategies are not mutually exclusive, and a literature review may combine elements of more than one approach. The choice of strategy depends on the research question, the nature of the literature available, and the goals of the review. Additionally, other strategies, such as integrative reviews or systematic reviews, may be employed depending on the specific requirements of the research.
The literature review format can vary depending on the specific publication guidelines. However, there are some common elements and structures that are often followed. Here is a general guideline for the format of a literature review: Introduction: Provide an overview of the topic. Define the scope and purpose of the literature review. State the research question or objective. Body: Organize the literature by themes, concepts, or chronology. Critically analyze and evaluate each source. Discuss the strengths and weaknesses of the studies. Highlight any methodological limitations or biases. Identify patterns, connections, or contradictions in the existing research. Conclusion: Summarize the key points discussed in the literature review. Highlight the research gap. Address the research question or objective stated in the introduction. Highlight the contributions of the review and suggest directions for future research.
Both annotated bibliographies and literature reviews involve the examination of scholarly sources. While annotated bibliographies focus on individual sources with brief annotations, literature reviews provide a more in-depth, integrated, and comprehensive analysis of existing literature on a specific topic. The key differences are as follows:
Annotated Bibliography | Literature Review | |
Purpose | List of citations of books, articles, and other sources with a brief description (annotation) of each source. | Comprehensive and critical analysis of existing literature on a specific topic. |
Focus | Summary and evaluation of each source, including its relevance, methodology, and key findings. | Provides an overview of the current state of knowledge on a particular subject and identifies gaps, trends, and patterns in existing literature. |
Structure | Each citation is followed by a concise paragraph (annotation) that describes the source’s content, methodology, and its contribution to the topic. | The literature review is organized thematically or chronologically and involves a synthesis of the findings from different sources to build a narrative or argument. |
Length | Typically 100-200 words | Length of literature review ranges from a few pages to several chapters |
Independence | Each source is treated separately, with less emphasis on synthesizing the information across sources. | The writer synthesizes information from multiple sources to present a cohesive overview of the topic. |
References
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Marco pautasso.
1 Centre for Functional and Evolutionary Ecology (CEFE), CNRS, Montpellier, France
2 Centre for Biodiversity Synthesis and Analysis (CESAB), FRB, Aix-en-Provence, France
Literature reviews are in great demand in most scientific fields. Their need stems from the ever-increasing output of scientific publications [1] . For example, compared to 1991, in 2008 three, eight, and forty times more papers were indexed in Web of Science on malaria, obesity, and biodiversity, respectively [2] . Given such mountains of papers, scientists cannot be expected to examine in detail every single new paper relevant to their interests [3] . Thus, it is both advantageous and necessary to rely on regular summaries of the recent literature. Although recognition for scientists mainly comes from primary research, timely literature reviews can lead to new synthetic insights and are often widely read [4] . For such summaries to be useful, however, they need to be compiled in a professional way [5] .
When starting from scratch, reviewing the literature can require a titanic amount of work. That is why researchers who have spent their career working on a certain research issue are in a perfect position to review that literature. Some graduate schools are now offering courses in reviewing the literature, given that most research students start their project by producing an overview of what has already been done on their research issue [6] . However, it is likely that most scientists have not thought in detail about how to approach and carry out a literature review.
Reviewing the literature requires the ability to juggle multiple tasks, from finding and evaluating relevant material to synthesising information from various sources, from critical thinking to paraphrasing, evaluating, and citation skills [7] . In this contribution, I share ten simple rules I learned working on about 25 literature reviews as a PhD and postdoctoral student. Ideas and insights also come from discussions with coauthors and colleagues, as well as feedback from reviewers and editors.
How to choose which topic to review? There are so many issues in contemporary science that you could spend a lifetime of attending conferences and reading the literature just pondering what to review. On the one hand, if you take several years to choose, several other people may have had the same idea in the meantime. On the other hand, only a well-considered topic is likely to lead to a brilliant literature review [8] . The topic must at least be:
Ideas for potential reviews may come from papers providing lists of key research questions to be answered [9] , but also from serendipitous moments during desultory reading and discussions. In addition to choosing your topic, you should also select a target audience. In many cases, the topic (e.g., web services in computational biology) will automatically define an audience (e.g., computational biologists), but that same topic may also be of interest to neighbouring fields (e.g., computer science, biology, etc.).
After having chosen your topic and audience, start by checking the literature and downloading relevant papers. Five pieces of advice here:
The chances are high that someone will already have published a literature review ( Figure 1 ), if not exactly on the issue you are planning to tackle, at least on a related topic. If there are already a few or several reviews of the literature on your issue, my advice is not to give up, but to carry on with your own literature review,
The bottom-right situation (many literature reviews but few research papers) is not just a theoretical situation; it applies, for example, to the study of the impacts of climate change on plant diseases, where there appear to be more literature reviews than research studies [33] .
When searching the literature for pertinent papers and reviews, the usual rules apply:
If you read the papers first, and only afterwards start writing the review, you will need a very good memory to remember who wrote what, and what your impressions and associations were while reading each single paper. My advice is, while reading, to start writing down interesting pieces of information, insights about how to organize the review, and thoughts on what to write. This way, by the time you have read the literature you selected, you will already have a rough draft of the review.
Of course, this draft will still need much rewriting, restructuring, and rethinking to obtain a text with a coherent argument [11] , but you will have avoided the danger posed by staring at a blank document. Be careful when taking notes to use quotation marks if you are provisionally copying verbatim from the literature. It is advisable then to reformulate such quotes with your own words in the final draft. It is important to be careful in noting the references already at this stage, so as to avoid misattributions. Using referencing software from the very beginning of your endeavour will save you time.
After having taken notes while reading the literature, you will have a rough idea of the amount of material available for the review. This is probably a good time to decide whether to go for a mini- or a full review. Some journals are now favouring the publication of rather short reviews focusing on the last few years, with a limit on the number of words and citations. A mini-review is not necessarily a minor review: it may well attract more attention from busy readers, although it will inevitably simplify some issues and leave out some relevant material due to space limitations. A full review will have the advantage of more freedom to cover in detail the complexities of a particular scientific development, but may then be left in the pile of the very important papers “to be read” by readers with little time to spare for major monographs.
There is probably a continuum between mini- and full reviews. The same point applies to the dichotomy of descriptive vs. integrative reviews. While descriptive reviews focus on the methodology, findings, and interpretation of each reviewed study, integrative reviews attempt to find common ideas and concepts from the reviewed material [12] . A similar distinction exists between narrative and systematic reviews: while narrative reviews are qualitative, systematic reviews attempt to test a hypothesis based on the published evidence, which is gathered using a predefined protocol to reduce bias [13] , [14] . When systematic reviews analyse quantitative results in a quantitative way, they become meta-analyses. The choice between different review types will have to be made on a case-by-case basis, depending not just on the nature of the material found and the preferences of the target journal(s), but also on the time available to write the review and the number of coauthors [15] .
Whether your plan is to write a mini- or a full review, it is good advice to keep it focused 16 , 17 . Including material just for the sake of it can easily lead to reviews that are trying to do too many things at once. The need to keep a review focused can be problematic for interdisciplinary reviews, where the aim is to bridge the gap between fields [18] . If you are writing a review on, for example, how epidemiological approaches are used in modelling the spread of ideas, you may be inclined to include material from both parent fields, epidemiology and the study of cultural diffusion. This may be necessary to some extent, but in this case a focused review would only deal in detail with those studies at the interface between epidemiology and the spread of ideas.
While focus is an important feature of a successful review, this requirement has to be balanced with the need to make the review relevant to a broad audience. This square may be circled by discussing the wider implications of the reviewed topic for other disciplines.
Reviewing the literature is not stamp collecting. A good review does not just summarize the literature, but discusses it critically, identifies methodological problems, and points out research gaps [19] . After having read a review of the literature, a reader should have a rough idea of:
It is challenging to achieve a successful review on all these fronts. A solution can be to involve a set of complementary coauthors: some people are excellent at mapping what has been achieved, some others are very good at identifying dark clouds on the horizon, and some have instead a knack at predicting where solutions are going to come from. If your journal club has exactly this sort of team, then you should definitely write a review of the literature! In addition to critical thinking, a literature review needs consistency, for example in the choice of passive vs. active voice and present vs. past tense.
Like a well-baked cake, a good review has a number of telling features: it is worth the reader's time, timely, systematic, well written, focused, and critical. It also needs a good structure. With reviews, the usual subdivision of research papers into introduction, methods, results, and discussion does not work or is rarely used. However, a general introduction of the context and, toward the end, a recapitulation of the main points covered and take-home messages make sense also in the case of reviews. For systematic reviews, there is a trend towards including information about how the literature was searched (database, keywords, time limits) [20] .
How can you organize the flow of the main body of the review so that the reader will be drawn into and guided through it? It is generally helpful to draw a conceptual scheme of the review, e.g., with mind-mapping techniques. Such diagrams can help recognize a logical way to order and link the various sections of a review [21] . This is the case not just at the writing stage, but also for readers if the diagram is included in the review as a figure. A careful selection of diagrams and figures relevant to the reviewed topic can be very helpful to structure the text too [22] .
Reviews of the literature are normally peer-reviewed in the same way as research papers, and rightly so [23] . As a rule, incorporating feedback from reviewers greatly helps improve a review draft. Having read the review with a fresh mind, reviewers may spot inaccuracies, inconsistencies, and ambiguities that had not been noticed by the writers due to rereading the typescript too many times. It is however advisable to reread the draft one more time before submission, as a last-minute correction of typos, leaps, and muddled sentences may enable the reviewers to focus on providing advice on the content rather than the form.
Feedback is vital to writing a good review, and should be sought from a variety of colleagues, so as to obtain a diversity of views on the draft. This may lead in some cases to conflicting views on the merits of the paper, and on how to improve it, but such a situation is better than the absence of feedback. A diversity of feedback perspectives on a literature review can help identify where the consensus view stands in the landscape of the current scientific understanding of an issue [24] .
In many cases, reviewers of the literature will have published studies relevant to the review they are writing. This could create a conflict of interest: how can reviewers report objectively on their own work [25] ? Some scientists may be overly enthusiastic about what they have published, and thus risk giving too much importance to their own findings in the review. However, bias could also occur in the other direction: some scientists may be unduly dismissive of their own achievements, so that they will tend to downplay their contribution (if any) to a field when reviewing it.
In general, a review of the literature should neither be a public relations brochure nor an exercise in competitive self-denial. If a reviewer is up to the job of producing a well-organized and methodical review, which flows well and provides a service to the readership, then it should be possible to be objective in reviewing one's own relevant findings. In reviews written by multiple authors, this may be achieved by assigning the review of the results of a coauthor to different coauthors.
Given the progressive acceleration in the publication of scientific papers, today's reviews of the literature need awareness not just of the overall direction and achievements of a field of inquiry, but also of the latest studies, so as not to become out-of-date before they have been published. Ideally, a literature review should not identify as a major research gap an issue that has just been addressed in a series of papers in press (the same applies, of course, to older, overlooked studies (“sleeping beauties” [26] )). This implies that literature reviewers would do well to keep an eye on electronic lists of papers in press, given that it can take months before these appear in scientific databases. Some reviews declare that they have scanned the literature up to a certain point in time, but given that peer review can be a rather lengthy process, a full search for newly appeared literature at the revision stage may be worthwhile. Assessing the contribution of papers that have just appeared is particularly challenging, because there is little perspective with which to gauge their significance and impact on further research and society.
Inevitably, new papers on the reviewed topic (including independently written literature reviews) will appear from all quarters after the review has been published, so that there may soon be the need for an updated review. But this is the nature of science [27] – [32] . I wish everybody good luck with writing a review of the literature.
Many thanks to M. Barbosa, K. Dehnen-Schmutz, T. Döring, D. Fontaneto, M. Garbelotto, O. Holdenrieder, M. Jeger, D. Lonsdale, A. MacLeod, P. Mills, M. Moslonka-Lefebvre, G. Stancanelli, P. Weisberg, and X. Xu for insights and discussions, and to P. Bourne, T. Matoni, and D. Smith for helpful comments on a previous draft.
This work was funded by the French Foundation for Research on Biodiversity (FRB) through its Centre for Synthesis and Analysis of Biodiversity data (CESAB), as part of the NETSEED research project. The funders had no role in the preparation of the manuscript.
A literature review basically has three functions:
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A systematic review of the key predictors of progression and mortality of rheumatoid arthritis-associated interstitial lung disease.
2. materials and methods, 2.1. eligibility, 2.2. search strategy, 2.3. data extraction, 3.1. selection of studies, 3.2. demographic features of eligible studies, 3.3. risk factors of lung function decline, 3.4. prognostic factors for all-cause mortality of ra-ild, 3.5. additional analysis, 4. discussion, 5. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.
Study | Design | Subjects (n) | Smoking (n, %) | UIP Pattern (n, %) | Frequency of Lung Function Decline (n, %) | All-Cause Mortality (n, %) |
---|---|---|---|---|---|---|
Akiyama et al., 2016 [ ] | Case/control retrospective (2008–2014) | 395 | 69 (20.3) | 78 (19.7) | 6 (1.5) | |
Franzen et al., 2016 [ ] | Observational retrospective (2013–2015) | 33 | 17 (51) | 6 (22) | ||
Md Yusof et al., 2017 [ ] | Observational retrospective (2004–2015) | 56 | 32 (57) | 20 (36) | 14 (25) | 9 (16) |
Mochizuki et al., 2018 [ ] | Observational retrospective, 47.8 months | 131 | ||||
Kim et al., 2010 [ ] | Observational retrospective (2001–2008) | 82 | 59 (72) | 20 (24) | 8 (9.7) | |
Zamora-Legoff et al., 2016 [ ] | Observational retrospective (1998–2014) | 167 | 105 (63) | 89 (53) | 33 (19) | |
Dawson et al., 2015 [ ] | Observational prospective, 2 years | 29 | 10 (34) | 10 (35) | 4 (14) | |
Solomon et al., 2015 [ ] | Observational retrospective (1995–2013) | 137 | 87 (64) | 108 (79) | 54 (39) | |
Dixon et al., 2010 [ ] | Observational registry-based cohort study, 3.8 years | 14.113 | 10799 (76) | 160 (1.13) | ||
Wolfe et al., 2007 [ ] | Case/control prospective, 3.6 years | 17498 | 100 (27) | |||
Kurata et al., 2019 [ ] | Observational retrospective (2008–2017) | 49 | 14 (35) | 6 (12) | ||
Chen et al., 2020 [ ] | Observational retrospective (2008–2017) | 241 | 88 (36) | 66 (27) | 39 (16) | |
Hyldgaard et al., 2017 [ ] | Case/control prospective (2004–2015) | 679 | 26 (3.8) | |||
Koduri et al., 2010 [ ] | Prospective cohort study (1986–1998) | 52 | 19 (36) | 39 (75) | ||
Tsuchiya et al., 2011 [ ] | Observational retrospective (1996–2006) | 144 | 54 (37.5) | 7 (5) | 71 (49) | |
Song et al., 2012 [ ] | Observational prospective (2002–2011) | 51 | 21 (41) | |||
Izuka et al., 2021 [ ] | Observational retrospective (2007–2019) | 165 | 73 (44) | 70 (42) | 30 (18) | 13 (8) |
Nurmi et al., 2017 [ ] | Observational retrospective (2000–2014) | 59 | 30 (51) | 31 (52) | 24 (41) | 27 (46) |
Oh et al., 2022 [ ] | Observational retrospective (1999–2015) | 144 | 63 (44) | 53 (37) | 44 (30) | |
Hozumi et al., 2022 [ ] | Observational retrospective (2007–2019) | 58 | 39 (67) | 34 (59) | 43 (74) | |
Cano-Jiménez et al., 2021 [ ] | Observational retrospective (2013–2018) | 106 | 63 (60) | 55(61) | 53 (50) | 18 (17) |
Jacob et al., 2018 [ ] | Observational retrospective (1995–2015) | 90 | 65 (72) | |||
Kelly et al., 2014 [ ] | Observational retrospective (1987–2012) | 230 | 135 (59) | 103 (65) | 90 (57) | |
Ng et al., 2022 [ ] | Observational retrospective (1997–2013) | 214 | 48 (22) | |||
Li et al., 2019 [ ] | Observational retrospective (2008–2017) | 278 | 106 (38) | 91 (33) | 83 (29) | 53 (69) |
Juge et al., 2023 [ ] | Observational retrospective (2013–2018) | 4330 | ||||
Brooks et al., 2022 [ ] | Prospective cohort study, 2 years | 227 | 192 (85) | 147 (65) | 108 (47) | |
Rojas-Serrano et al., 2022 [ ] | Observational prospective | 37 | ||||
Kim et al., 2020 [ ] | Observational retrospective (1995–2018) | 84 | 37 (44) | 34 (40) | 33 (39) | |
Lee et al., 2016 [ ] | Retrospective cohort study | 62 | 62 (100) | |||
Avouac et al.2020 [ ] | Observational prospective study | 147 | 52 (35) | 21 (14) | 7 (4.7) | |
Ito et al., 2017 [ ] | Observational retrospective (2007–2016) | 65 | 16 (24) | |||
Font et al., 2017 [ ] | Longitudinal prospective (2007–2017) | 37 | 26(72) | 24(66) | 7 (19) | |
Nieto et al., 2021 [ ] | Longitudinal prospective (2005–2018) | 47 | 25 (54) | 26 (55) | 10 (21) | 16 (34) |
Yang et al., 2019 [ ] | Longitudinal prospective (1991–2011) | 77 | 4 (5) | 32 (48) | 27 (36) | |
Mena-Vázquez et al., 2024 [ ] | Observational prospective (2015–2023) | 148 | 13 (18) | 46 (66) | 21 (30) | 1 (1.4) |
Kim et al., 2024 [ ] | Observational retrospective (1995–2018) | 313 | 139 (45) | 202 (65) | 125 (40) | |
Chen et al., 2022 [ ] | Observational prospective, 5 years | 60 | 3 (16) | 19 (49) | 4 (7) | |
Kelly et al., 2021 [ ] | Observational retrospective (1990–2015) | 290 | 174 (60) | 200 (69) | 75 (28) | |
Venerito et al., 2022 [ ] | Observational retrospective (2021–2022) | 30 | 18 (60) | 13 (43) | ||
Yamakawa et al., 2019 [ ] | Observational retrospective (2012–2017) | 96 | 45 (47) | 21 (20) | 11 (11) | 25 (26) |
Ekici et al., 2021 [ ] | Observational retrospective (2010–2018) | 156 | 67 (42) | 74 (47) | 40 (26) | |
Kakutani et al., 2020 [ ] | Observational retrospective (2009–2014) | 261/2702 | 120 (46) | 120 (46) | 19 (7) | |
Wang et al., 2019 [ ] | Observational retrospective (2016–2019) | 96 | 9 (20) | 18 (40) | 25 (56) | 4 (9) |
Tanaka et al., 2021 [ ] | Observational retrospective (2010–2019) | 125 | 59 (50) | 32 (25) | 37 (29) | |
Kwon et al., 2022 [ ] | Observational retrospective (2016–2022) | 310 | 87 (28) | 89 (29) | ||
Kang et al., 2020 [ ] | Retrospective cohort (2006–2015) | 1999 | 759 (38) | 415 (21) | ||
Farquhar et al., 2024 [ ] | Observational retrospective (2006–2008, 2011–2013) | 100 | 64 (64) | 38 (43) | 68 (73) | 26 (26) |
Marcoux et al., 2023 [ ] | Observational prospective (2015–2018) | 181 | 112 (70) | 66 (73) | 39 (24) | |
Tyker et al., 2021 [ ] | Observational retrospective (2006–2019) | 70 | 45 (64) | 47 (67) | 29 (70) |
Category | Potential Risk Factors | References | Effect Estimate |
---|---|---|---|
Demographic features | Age * | [ , , ] | OR 0.55–2.91, AUC 0.74 |
CDAI score * | [ ] | OR 4.7 | |
Laboratory findings | ACPA ** | [ ] | HR 3.94 |
KL-6 * | [ , , , ] | OR 1.00–72.7 | |
MMP13 ** | [ ] | AUC 0.71 | |
CXCL11/I-TAC ** | [ ] | AUC 0.67 | |
MUC5B mutation ** | [ ] | HR 2.30 | |
Pulmonary function | FVC% pred ** | [ ] | HR 3.42 |
DLCO% pred ** | [ ] | HR 1.72 | |
Underlying radiological features | UIP pattern on HRCT * | [ , ] | OR 2.29–4.11 |
Pre-treatment | Corticosteroid * | [ ] | HR 15.0 |
Nonbiologic DMARDs * | [ , ] | OR 1.75–.75 | |
Biologic DMARDs ** | [ , ] | HR 0.44–2.33 |
Category | Potential Risk Factors | References | Effect Estimate |
---|---|---|---|
Demographic features | Age ** | [ , , , , , , , , , , , ] | HR 1.04–4.8 |
Male gender ** | [ , , ] | HR 2.83–14.5 | |
Female gender ** | [ ] | HR 3.6 | |
Smoking history ** | [ , ] | HR 2.58–3.17 | |
Low socioeconomic status * | [ ] | HR 2.07 | |
The onset of ILD before RA onset | [ ] | HR 8.4 | |
Disease activity | DAS28 score ** | [ ] | HR 1.21–1.43 |
CDAI score * | [ ] | HR 1.07 | |
MDHAQ score ** | [ ] | HR 1.85 | |
Pain VAS * | [ ] | HR 1.01 | |
Patient global assessment ** | [ ] | HR 1.16 | |
Laboratory findings | RF * | [ , , ] | HR 1.00–2.08 |
ESR ** | [ , , ] | HR 1.01–1.15 | |
CRP ** | [ ] | HR 1.12 (1.06–1.18) | |
LDH * | [ ] | HR 1.05 | |
KL-6 ** | [ , ] | HR 1.00–3.23 | |
IL-6 ** | [ ] | HR 1.04 | |
SP-D * | [ ] | HR 1.0 | |
Pulmonary function | DLCO % pred ** | [ , , , ] | HR 0.97–1.77 |
FVC % pred ** | [ , , , , , ] | HR 0.97–4.43 | |
TLCO % pred ** | [ ] | HR 0.97 | |
PaO /FiO * | [ ] | HR 0.94 | |
Underlying radiological features | ILD extent ** | [ , , , ] | HR 1.03–4.47 |
UIP ** | [ , , , , , , ] | HR 2.44–5.84 | |
Honeycombing * | [ ] | HR 2.49 | |
Radiomics * | [ ] | HR 9.35 | |
DAD * | [ ] | HR 2.88 | |
Emphysema * | [ , ] | HR 3.43–6.84 | |
Comorbidities | COPD ** | [ ] | HR 2.12 |
Diabetes mellitus ** | [ ] | HR 1.09 | |
Pre-treatment | Corticosteroid ** | [ ] | HR 1.09 |
Nonbiologic DMARDs * | [ , ] | HR 0.16–5.53 | |
Biologic DMARDs ** | [ ] | HR 0.44–2.33 | |
Acute exacerbations of ILD ** | [ ] | HR 1.12–3.19 |
Click here to enlarge figure
Potential Risk Factors | Study | Effect Estimate |
---|---|---|
Age at ILD diagnosis ** | 44, 70 | HR 2.18; OR 1.7 |
Male gender ** | 70 | OR 2.2 |
Smoking history *** | 37, 70 | OR 1.7–6.13 |
DAS28 *** | 37 | OR 1.71 |
Arthritis onset ** | 44 | HR 1.87 |
KL-6 ** | 74 | HR 3.37 |
DLCO ** | 53 | OR 3.02 |
UIP pattern ** | 53 | OR 3.47 |
Combined pulmonary fibrosis and emphysema ** | 59 | OR 6.12 |
Preexisting rheumatic airway disease ** | 95 | OR 7.40 |
Prognostic Factor | References | Effect Estimate |
---|---|---|
Age ** | [ , , , , , , , , , ] | HR 1.04–5.02 |
Male gender ** | [ ] | OR 2.5–18.13 |
Female gender ** | [ ] | HR 6.8 |
Smoking history * | [ ] | HR 1.06–3.89 |
Disease duration of RA ** | [ ] | HR 1.3 |
ESR ** | [ ] | HR 5.35 |
HAQ disability ** | [ ] | OR 2.3 |
Steinbrocker class 3 or 4 ** | [ ] | HR 2.1 |
FVC% pred * | [ ] | HR 2.52 |
DLCO *** | [ , ] | HR 0.85–0.97 |
TLCO ** | [ ] | HR 0.98 |
Final oxygen saturation in the 6MWT ** | [ ] | HR 0.62 |
UIP pattern ** | [ , , , , ] | HR 2.3–10.3 |
Non-UIP pattern ** | [ ] | HR 4.9 |
ILD extent *** | [ , ] | HR 2.40–9.01 |
Radiological honeycombing ** | [ ] | HR 3.69 |
Combined pulmonary fibrosis and emphysema ** | [ ] | HR 2.16 |
Pleural effusion ** | [ ] | HR 14.4 |
Corticosteroid * | [ ] | HR 2.5 |
Immunosuppressive agents ** | [ ] | HR 3.0 |
Withdrawal of MTX or LFN after ILD diagnosis ** | [ ] | HR 2.18 |
Diagnostic delay of ILD ** | [ ] | HR 1.11 |
PM * | [ ] | HR 1.67 |
History of acute ILD exacerbations *** | [ , ] | HR 2.42–6.48 |
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
Groseanu, L.; Niță, C. A Systematic Review of the Key Predictors of Progression and Mortality of Rheumatoid Arthritis-Associated Interstitial Lung Disease. Diagnostics 2024 , 14 , 1890. https://doi.org/10.3390/diagnostics14171890
Groseanu L, Niță C. A Systematic Review of the Key Predictors of Progression and Mortality of Rheumatoid Arthritis-Associated Interstitial Lung Disease. Diagnostics . 2024; 14(17):1890. https://doi.org/10.3390/diagnostics14171890
Groseanu, Laura, and Cristina Niță. 2024. "A Systematic Review of the Key Predictors of Progression and Mortality of Rheumatoid Arthritis-Associated Interstitial Lung Disease" Diagnostics 14, no. 17: 1890. https://doi.org/10.3390/diagnostics14171890
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In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
Given two sets \(A\) and \(B\) a set with elements that are ordered pairs \((x,y)\) where \(x\) is an element of \(A\) and \(y\) is an element of \(B,\) is a relation from \(A\) to \(B\). A relation from \(A\) to \(B\) defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input ; the element of the second set is called the output . Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.
A function \(f\) consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain of the function. The set of outputs is called the range of the function .
For example, consider the function \(f\), where the domain is the set of all real numbers and the rule is to square the input. Then, the input \(x=3\) is assigned to the output \(3^2=9\).
Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers.
For a general function \(f\) with domain \(D\), we often use \(x\) to denote the input and \(y\) to denote the output associated with \(x\). When doing so, we refer to \(x\) as the independent variable and \(y\) as the dependent variable , because it depends on \(x\). Using function notation, we write \(y=f(x)\), and we read this equation as “\(y\) equals \(f\) of \(x.”\) For the squaring function described earlier, we write \(f(x)=x^2\).
The concept of a function can be visualized using Figures \(\PageIndex{1}\) - \(\PageIndex{3}\).
We can also visualize a function by plotting points \((x,y)\) in the coordinate plane where \(y=f(x)\). The graph of a function is the set of all these points. For example, consider the function \(f\), where the domain is the set \(D=\{1,2,3\}\) and the rule is \(f(x)=3−x\). In Figure \(\PageIndex{4}\), we plot a graph of this function.
Every function has a domain. However, sometimes a function is described by an equation, as in \(f(x)=x^2\), with no specific domain given. In this case, the domain is taken to be the set of all real numbers \(x\) for which \(f(x)\) is a real number. For example, since any real number can be squared, if no other domain is specified, we consider the domain of \(f(x)=x^2\) to be the set of all real numbers. On the other hand, the square root function \(f(x)=\sqrt{x}\) only gives a real output if \(x\) is nonnegative. Therefore, the domain of the function \(f(x)=\sqrt{x}\) is the set of nonnegative real numbers, sometimes called the natural domain.
For the functions \(f(x)=x^2\) and \(f(x)=\sqrt{x}\), the domains are sets with an infinite number of elements. Clearly we cannot list all these elements. When describing a set with an infinite number of elements, it is often helpful to use set-builder or interval notation. When using set-builder notation to describe a subset of all real numbers, denoted \(R\), we write
\[\{x\,|\,\textit{x has some property}\}. \nonumber \]
We read this as the set of real numbers \(x\) such that \(x\) has some property. For example, if we were interested in the set of real numbers that are greater than one but less than five, we could denote this set using set-builder notation by writing
\[\{x\,|\,1<x<5\}.\nonumber \]
A set such as this, which contains all numbers greater than \(a\) and less than \(b,\) can also be denoted using the interval notation \((a,b)\). Therefore,
\[(1,5)=\{x\,|\,1<x<5\}.\nonumber \]
The numbers \(1\) and \(5\) are called the endpoints of this set. If we want to consider the set that includes the endpoints, we would denote this set by writing
\[[1,5]=\{x\,|\,1 \le x \le 5\}.\nonumber \]
We can use similar notation if we want to include one of the endpoints, but not the other. To denote the set of nonnegative real numbers, we would use the set-builder notation
\[\{x\,|\,x\ge 0\}.\nonumber \]
The smallest number in this set is zero, but this set does not have a largest number. Using interval notation, we would use the symbol \(∞,\) which refers to positive infinity, and we would write the set as
\[[0,∞)=\{x\,|\,x\ge 0\}.\nonumber \]
It is important to note that \(∞\) is not a real number. It is used symbolically here to indicate that this set includes all real numbers greater than or equal to zero. Similarly, if we wanted to describe the set of all nonpositive numbers, we could write
\[(−∞,0]=\{x\,|\,x≤0\}.\nonumber \]
Here, the notation \(−∞\) refers to negative infinity, and it indicates that we are including all numbers less than or equal to zero, no matter how small. The set
\[(−∞,∞)=\{\textit{x} \,|\, \textit{x is any real number}\}\nonumber \]
refers to the set of all real numbers. Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions. For example, suppose we want to define a function \(f\) with a domain that is the set of all real numbers such that \(f(x)=3x+1\) for \(x≥2\) and \(f(x)=x^2\) for \( x<2\). We denote this function by writing
\[f(x)=\begin{cases} 3x+1, & \text{if } x≥2 \\ x^2, & \text{if } x<2 \end{cases}\nonumber \]
When evaluating this function for an input \(x\), the equation to use depends on whether \(x≥2\) or \(x<2\). For example, since \(5>2\), we use the fact that \(f(x)=3x+1\) for \(x≥2\) and see that \(f(5)=3(5)+1=16\). On the other hand, for \(x=−1\), we use the fact that \(f(x)=x^2\) for \(x<2\) and see that \(f(−1)=1\).
For the function \(f(x)=3x^2+2x−1\), evaluate:
Substitute the given value for \(x\) in the formula for \(f(x)\).
For \(f(x)=x^2−3x+5\), evaluate \(f(1)\) and \(f(a+h)\).
Substitute \(1\) and \(a+h\) for \(x\) in the formula for \(f(x)\).
\(f(1)=3 \) and \(f(a+h)=a^2+2ah+h^2−3a−3h+5\)
For each of the following functions, determine the i. domain and ii. range.
a. Consider \(f(x)=(x−4)^2+5.\)
1.Since \(f(x)=(x−4)^2+5\) is a real number for any real number \(x\), the domain of \(f\) is the interval \((−∞,∞)\).
2. Since \((x−4)^2≥0\), we know \(f(x)=(x−4)^2+5≥5\). Therefore, the range must be a subset of \(\{y\,|\,y≥5\}.\) To show that every element in this set is in the range, we need to show that for a given \(y\) in that set, there is a real number \(x\) such that \(f(x)=(x−4)^2+5=y\). Solving this equation for \(x,\) we see that we need \(x\) such that
\((x−4)^2=y−5.\)
This equation is satisfied as long as there exists a real number \(x\) such that
\(x−4=±\sqrt{y−5}\)
Since \(y≥5\), the square root is well-defined. We conclude that for \(x=4±\sqrt{y−5},\) \(f(x)=y,\) and therefore the range is \(\{y\,|\,y≥5 \}.\)
b. Consider \(f(x)=\sqrt{3x+2}−1\).
1.To find the domain of \(f\), we need the expression \(3x+2≥0\). Solving this inequality, we conclude that the domain is \(\{x\,|\,x≥−2/3\}.\)
2.To find the range of \(f\), we note that since \(\sqrt{3x+2}≥0,\) \(f(x)=\sqrt{3x+2}−1≥−1\). Therefore, the range of \(f\) must be a subset of the set \(\{y\,|\,y≥−1\}\). To show that every element in this set is in the range of \(f\), we need to show that for all \(y\) in this set, there exists a real number \(x\) in the domain such that \(f(x)=y.\) Let \(y≥−1.\) Then, \(f(x)=y\) if and only if
\(\sqrt{3x+2}−1=y.\)
Solving this equation for \(x,\) we see that \(x\) must solve the equation
\(\sqrt{3x+2}=y+1.\)
Since \(y≥−1\), such an \(x\) could exist. Squaring both sides of this equation, we have \(3x+2=(y+1)^2.\)
Therefore, we need
\(3x=(y+1)^2−2,\)
which implies
\(x=\frac{1}{3}(y+1)^2−\frac{2}{3}.\)
We just need to verify that \(x\) is in the domain of \(f\). Since the domain of \(f\) consists of all real numbers greater than or equal to \(\frac{−2}{3}\), and
\(\frac{1}{3}(y+1)^2-\frac{2}{3}≥−\frac{2}{3},\)
there does exist an \(x\) in the domain of \(f\). We conclude that the range of \(f\) is \(\{y\,|\,y≥−1\}.\)
c. Consider \(f(x)=\dfrac{3}{x−2}.\)
1.Since \(3/(x−2)\) is defined when the denominator is nonzero, the domain is \(\{x\,|\,x≠2\}.\)
2.To find the range of \(f,\) we need to find the values of \(y\) such that there exists a real number \(x\) in the domain with the property that
\(\dfrac{3}{x−2}=y.\)
Solving this equation for \(x,\) we find that
\(x=\dfrac{3}{y}+2.\)
Therefore, as long as \(y≠0\), there exists a real number \(x\) in the domain such that \(f(x)=y\). Thus, the range is \(\{y\,|\,y≠0\}.\)
Find the domain and range for \(f(x)=\sqrt{4−2x}+5.\)
Use \(4−2x≥0\).
Domain = \(\{x\,|\,x≤2\}\) and range = \(\{y\,|\,y≥5\}\)
Typically, a function is represented using one or more of the following tools:
We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.
Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable \(x\) be the time after midnight, measured in hours, and the output variable \(y\) be the temperature \(x\) hours after midnight, measured in degrees Fahrenheit. We record our data in Table \(\PageIndex{1}\).
0 | 58 | 12 | 84 |
1 | 54 | 13 | 85 |
2 | 53 | 14 | 85 |
3 | 52 | 15 | 83 |
4 | 52 | 16 | 82 |
5 | 55 | 17 | 80 |
6 | 60 | 18 | 77 |
7 | 64 | 19 | 74 |
8 | 72 | 20 | 69 |
9 | 75 | 21 | 65 |
10 | 78 | 22 | 60 |
11 | 80 | 23 | 58 |
We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.
Given a function \(f\) described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table \(\PageIndex{1}\) can give us a better idea of their fluctuation throughout the day. Figure \(\PageIndex{5}\) shows the plot of the temperature function.
From the points plotted on the graph in Figure \(\PageIndex{5}\), we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in Figure \(\PageIndex{6}\).
Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius \(r\) is given by the formula \(A(r)=πr^2\). When an object is thrown upward from the ground with an initial velocity \(v_{0}\) ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula \(s(t)=−16t^2+v_{0}t\). When \(P\) dollars are invested in an account at an annual interest rate \(r\) compounded continuously, the amount of money after \(t\) years is given by the formula \(A(t)=Pe^{rt}\). Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.
Given an algebraic formula for a function \(f\), the graph of \(f\) is the set of points \((x,f(x))\), where \(x\) is in the domain of \(f\) and \(f(x)\) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of \(f\) consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.
When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of \(x\) where \(f(x)=0\) are called the zeros of a function. For example, the zeros of \(f(x)=x^2−4\) are \(x=±2\). The zeros determine where the graph of \(f\) intersects the \(x\)-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the \(x\)-axis, or it may intersect multiple (or even infinitely many) times.
Another point of interest is the \(y\) -intercept, if it exists. The \(y\)-intercept is given by \((0,f(0))\).
Since a function has exactly one output for each input, the graph of a function can have, at most, one \(y\)-intercept. If \(x=0\) is in the domain of a function \(f,\) then \(f\) has exactly one \(y\)-intercept. If \(x=0\) is not in the domain of \(f,\) then \(f\) has no \(y\)-intercept. Similarly, for any real number \(c,\) if \(c\) is in the domain of \(f\), there is exactly one output \(f(c),\) and the line \(x=c\) intersects the graph of \(f\) exactly once. On the other hand, if \(c\) is not in the domain of \(f,\) \(f(c)\) is not defined and the line \(x=c\) does not intersect the graph of \(f\). This property is summarized in the vertical line test.
Given a function \(f\), every vertical line that may be drawn intersects the graph of \(f\) no more than once. If any vertical line intersects a set of points more than once, the set of points does not represent a function.
We can use this test to determine whether a set of plotted points represents the graph of a function (Figure \(\PageIndex{7}\)).
Consider the function \(f(x)=−4x+2.\)
1.To find the zeros, solve \(f(x)=−4x+2=0\). We discover that \(f\) has one zero at \(x=1/2\).
2. The \(y\)-intercept is given by \((0,f(0))=(0,2).\)
3. Given that \(f\) is a linear function of the form \(f(x)=mx+b\) that passes through the points \((1/2,0)\) and \((0,2)\), we can sketch the graph of \(f\) (Figure \(\PageIndex{8}\)).
Consider the function \(f(x)=\sqrt{x+3}+1\).
1.To find the zeros, solve \(\sqrt{x+3}+1=0\). This equation implies \(\sqrt{x+3}=−1\). Since \(\sqrt{x+3}≥0\) for all \(x\), this equation has no solutions, and therefore \(f\) has no zeros.
2.The \(y\)-intercept is given by \((0,f(0))=(0,\sqrt{3}+1)\).
3.To graph this function, we make a table of values. Since we need \(x+3≥0\), we need to choose values of \(x≥−3\). We choose values that make the square-root function easy to evaluate.
\(x\) | -3 | -2 | 1 |
---|---|---|---|
\(f(x)\) | 1 | 2 | 3 |
Making use of the table and knowing that, since the function is a square root, the graph of \(f\) should be similar to the graph of \(y=\sqrt{x}\), we sketch the graph (Figure \(\PageIndex{9}\)).
Find the zeros of \(f(x)=x^3−5x^2+6x.\)
Factor the polynomial.
\(x=0,2,3\)
If a ball is dropped from a height of 100 ft, its height s at time \(t\) is given by the function \(s(t)=−16t^2+100\), where s is measured in feet and \(t\) is measured in seconds. The domain is restricted to the interval \([0,c],\) where \(t=0\) is the time when the ball is dropped and \(t=c\) is the time when the ball hits the ground.
\(t\) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 |
\(s(t)\) | 100 | 96 | 84 | 64 | 36 | 0 |
Since the ball hits the ground when \(t=2.5\), the domain of this function is the interval \([0,2.5]\).
We say that a function \(f\) is increasing on the interval \(I\) if for all \(x_{1},\, x_{2}∈I,\)
\(f(x_{1})≤f(x_{2})\) when \(x_{1}<x_{2}.\)
We say \(f\) is strictly increasing on the interval \(I\) if for all \(x_{1},x_{2}∈I,\)
\(f(x_{1})<f(x_{2})\) when \(x_{1}<x_{2}.\)
We say that a function \(f\) is decreasing on the interval \(I\) if for all \(x_{1},x_{2}∈I,\)
\(f(x_{1})≥f(x_{2})\) if \(x_{1}<x_{2}.\)
We say that a function \(f\) is strictly decreasing on the interval \(I\) if for all \(x_{1},x_{2}∈I\),
\(f(x_{1})>f(x_{2})\) if \(x_{1}<x_{2}.\)
For example, the function \(f(x)=3x\) is increasing on the interval \((−∞,∞)\) because \(3x_{1}<3x_{2}\) whenever \(x_{1}<x_{2}\). On the other hand, the function \(f(x)=−x^3\) is decreasing on the interval \((−∞,∞)\) because \(−x^3_{1}>−x^3_{2}\) whenever \(x_{1}<x_{2}\) (Figure \(\PageIndex{10}\)).
Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture \(x\) items is described by the function \(C(x)\) and the revenue created by the sale of \(x\) items is described by the function \(R(x)\), then the profit on the manufacture and sale of \(x\) items is defined as \(P(x)=R(x)−C(x)\). Using the difference between two functions, we created a new function.
Alternatively, we can create a new function by composing two functions. For example, given the functions \(f(x)=x^2\) and \(g(x)=3x+1\), the composite function \(f∘g\) is defined such that
\[(f∘g)(x)=f(g(x))=(g(x))^2=(3x+1)^2. \nonumber \]
The composite function \(g∘f\) is defined such that
\[(g∘f)(x)=g(f(x))=3f(x)+1=3x^2+1. \nonumber \]
Note that these two new functions are different from each other.
To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions \(f\) and \(g\), we can define four new functions:
\((f+g)(x)=f(x)+g(x)\) | Sum |
\((f−g)(x)=f(x)−g(x)\) | Difference |
\((f·g)(x)=f(x)g(x)\) | Product |
\((\frac{f}{g})(x)=\frac{f(x)}{g(x)}\) for\(g(x)≠0\) | Quotient |
Given the functions \(f(x)=2x−3\) and \(g(x)=x^2−1\), find each of the following functions and state its domain.
1. \((f+g)(x)=(2x−3)+(x^2−1)=x^2+2x−4.\)
The domain of this function is the interval \((−∞,∞)\).
2.\((f−g)(x)=(2x−3)−(x^2−1)=−x^2+2x−2.\)
3. \((f·g)(x)=(2x−3)(x^2−1)=2x^3−3x^2−2x+3.\)
4. \(\left(\dfrac{f}{g}\right)(x)=\dfrac{2x−3}{x^2−1}\).
The domain of this function is \(\{x\,|\,x≠±1\}.\)
For \(f(x)=x^2+3\) and \(g(x)=2x−5\), find \((f/g)(x)\) and state its domain.
The new function \((f/g)(x)\) is a quotient of two functions. For what values of \(x\) is the denominator zero?
\(\left(\dfrac{f}{g}\right)(x)=\frac{x^2+3}{2x−5}.\) The domain is \(\{x\,|\,x≠\frac{5}{2}\}.\)
When we compose functions, we take a function of a function. For example, suppose the temperature \(T\) on a given day is described as a function of time \(t\) (measured in hours after midnight) as in Table \(\PageIndex{1}\). Suppose the cost \(C\), to heat or cool a building for 1 hour, can be described as a function of the temperature \(T\). Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating \(C(T(t))\). We have defined a new function, denoted \(C∘T\), which is defined such that \((C∘T)(t)=C(T(t))\) for all \(t\) in the domain of \(T\). This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function \((C∘T)(t)\). It does not make sense to consider \((T∘C)(t)\), because temperature is not a function of cost.
Consider the function \(f\) with domain \(A\) and range \(B\), and the function \(g\) with domain \(D\) and range \(E\). If \(B\) is a subset of \(D\), then the composite function \((g∘f)(x)\) is the function with domain \(A\) such that
\[(g∘f)(x)=g(f(x)) \nonumber \]
A composite function \(g∘f\) can be viewed in two steps. First, the function \(f\) maps each input \(x\) in the domain of \(f\) to its output \(f(x)\) in the range of \(f\). Second, since the range of \(f\) is a subset of the domain of \(g\), the output \(f(x)\) is an element in the domain of \(g\), and therefore it is mapped to an output \(g(f(x))\) in the range of \(g\). In Figure \(\PageIndex{11}\), we see a visual image of a composite function.
Consider the functions \(f(x)=x^2+1\) and \(g(x)=1/x\).
1. We can find the formula for \((g∘f)(x)\) in two different ways. We could write
\((g∘f)(x)=g(f(x))=g(x^2+1)=\dfrac{1}{x^2+1}\).
Alternatively, we could write
\((g∘f)(x)=g(f(x))=\dfrac{1}{f(x)}=\dfrac{1}{x^2+1}.\)
Since \(x^2+1≠0\) for all real numbers \(x,\) the domain of \((g∘f)(x)\) is the set of all real numbers. Since \(0<1/(x^2+1)≤1\), the range is, at most, the interval \((0,1]\). To show that the range is this entire interval, we let \(y=1/(x^2+1)\) and solve this equation for \(x\) to show that for all \(y\) in the interval \((0,1]\), there exists a real number \(x\) such that \(y=1/(x^2+1)\). Solving this equation for \(x,\) we see that \(x^2+1=1/y\), which implies that
\(x=±\sqrt{\frac{1}{y}−1}\)
If \(y\) is in the interval \((0,1]\), the expression under the radical is nonnegative, and therefore there exists a real number \(x\) such that \(1/(x^2+1)=y\). We conclude that the range of \(g∘f\) is the interval \((0,1].\)
2. \((g∘f)(4)=g(f(4))=g(4^2+1)=g(17)=\frac{1}{17}\)
\((g∘f)(−\frac{1}{2})=g(f(−\frac{1}{2}))=g((−\frac{1}{2})^2+1)=g(\frac{5}{4})=\frac{4}{5}\)
3. We can find a formula for \((f∘g)(x)\) in two ways. First, we could write
\((f∘g)(x)=f(g(x))=f(\frac{1}{x})=(\frac{1}{x})^2+1.\)
\((f∘g)(x)=f(g(x))=(g(x))^2+1=(\frac{1}{x})^2+1.\)
The domain of \(f∘g\) is the set of all real numbers \(x\) such that \(x≠0\). To find the range of \(f,\) we need to find all values \(y\) for which there exists a real number \(x≠0\) such that
\(\left(\dfrac{1}{x}\right)^2+1=y.\)
Solving this equation for \(x,\) we see that we need \(x\) to satisfy
\(\left(\dfrac{1}{x}\right)^2=y−1,\)
which simplifies to
\(\dfrac{1}{x}=±\sqrt{y−1}\)
Finally, we obtain
\(x=±\dfrac{1}{\sqrt{y−1}}.\)
Since \(1/\sqrt{y−1}\) is a real number if and only if \(y>1,\) the range of \(f\) is the set \(\{y\,|\,y≥1\}.\)
4.\((f∘g)(4)=f(g(4))=f(\frac{1}{4})=(\frac{1}{4})^2+1=\frac{17}{16}\)
\((f∘g)(−\frac{1}{2})=f(g(−\frac{1}{2}))=f(−2)=(−2)^2+1=5\)
In Example \(\PageIndex{7}\) , we can see that \((f∘g)(x)≠(g∘f)(x)\). This tells us, in general terms, that the order in which we compose functions matters.
Let \(f(x)=2−5x\). Let \(g(x)=\sqrt{x}.\) Find \((f∘g)(x)\).
\((f∘g)(x)=2−5\sqrt{x}.\)
Consider the functions \(f\) and \(g\) described by
\(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|---|---|
\(f(x)\) | 0 | 4 | 2 | 4 | -2 | 0 | -2 | 4 |
\(x\) | -4 | -2 | 0 | 2 | 4 |
---|---|---|---|---|---|
\(g(x)\) | 1 | 0 | 3 | 0 | 5 |
1. \((g∘f)(3)=g(f(3))=g(−2)=0\)
\((g∘f)(0)=g(4)=5\)
2.The domain of \(g∘f\) is the set \(\{−3,−2,−1,0,1,2,3,4\}.\) Since the range of \(f\) is the set \(\{−2,0,2,4\},\) the range of \(g∘f\) is the set \(\{0,3,5\}.\)
3. \((f∘f)(3)=f(f(3))=f(−2)=4\)
\((f∘f)(1)=f(f(1))=f(−2)=4\)
4.The domain of \(f∘f\) is the set \(\{−3,−2,−1,0,1,2,3,4\}.\) Since the range of \(f\) is the set \(\{−2,0,2,4\},\) the range of \(f∘f\) is the set \(\{0,4\}.\)
A store is advertising a sale of 20% off all merchandise. Caroline has a coupon that entitles her to an additional 15% off any item, including sale merchandise. If Caroline decides to purchase an item with an original price of \(x\) dollars, how much will she end up paying if she applies her coupon to the sale price? Solve this problem by using a composite function.
Since the sale price is 20% off the original price, if an item is \(x\) dollars, its sale price is given by \(f(x)=0.80x\). Since the coupon entitles an individual to 15% off the price of any item, if an item is \(y\) dollars, the price, after applying the coupon, is given by g(y)=0.85y. Therefore, if the price is originally \(x\) dollars, its sale price will be \(f(x)=0.80x\) and then its final price after the coupon will be \(g(f(x))=0.85(0.80x)=0.68x\).
If items are on sale for 10% off their original price, and a customer has a coupon for an additional 30% off, what will be the final price for an item that is originally \(x\) dollars, after applying the coupon to the sale price?
The sale price of an item with an original price of \(x\) dollars is \(f(x)=0.90x\). The coupon price for an item that is \(y\) dollars is \(g(y)=0.70y\).
\((g∘f)(x)=0.63x\)
The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function \(f(x)=x^4−2x^2−3\) shown in Figure \(\PageIndex{12a}\). If we take the part of the curve that lies to the right of the \(y\)-axis and flip it over the \(y\)-axis, it lays exactly on top of the curve to the left of the \(y\)-axis. In this case, we say the function has symmetry about the \(y\)-axis . On the other hand, consider the function \(f(x)=x^3−4x\) shown in Figure \(\PageIndex{12b}\). If we take the graph and rotate it \(180°\) about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin .
If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function \(f\) has symmetry? Looking at Figure \(\PageIndex{12a}\) again, we see that since \(f\) is symmetric about the \(y\)-axis, if the point \((x,y)\) is on the graph, the point \((−x,y)\) is on the graph. In other words, \(f(−x)=f(x)\). If a function \(f\) has this property, we say \(f\) is an even function, which has symmetry about the \(y\)-axis. For example, \(f(x)=x^2\) is even because
\(f(−x)=(−x)^2=x^2=f(x).\)
In contrast, looking at Figure \(\PageIndex{12b}\) again, if a function \(f\) is symmetric about the origin, then whenever the point \((x,y)\) is on the graph, the point \((−x,−y)\) is also on the graph. In other words, \(f(−x)=−f(x)\). If \(f\) has this property, we say \(f\) is an odd function, which has symmetry about the origin. For example, \(f(x)=x^3\) is odd because
\(f(−x)=(−x)^3=−x^3=−f(x).\)
Determine whether each of the following functions is even, odd, or neither.
To determine whether a function is even or odd, we evaluate \(f(−x)\) and compare it to \(f(x)\) and \(−f(x)\).
1. \(f(−x)=−5(−x)^4+7(−x)^2−2=−5x^4+7x^2−2=f(x).\) Therefore, \(f\) is even.
2.\(f(−x)=2(−x)^5−4(−x)+5=−2x^5+4x+5.\) Now, \(f(−x)≠f(x).\) Furthermore, noting that \(−f(x)=−2x^5+4x−5\), we see that \(f(−x)≠−f(x)\). Therefore, \(f\) is neither even nor odd.
3.\(f(−x)=3(−x)/((−x)2+1)\)\(=−3x/(x^2+1)=\)\(−[3x/(x^2+1)]=−f(x).\) Therefore, \(f\) is odd.
Determine whether \(f(x)=4x^3−5x\) is even, odd, or neither.
Compare \(f(−x)\) with \(f(x)\) and \(−f(x)\).
\(f(x)\) is odd.
One symmetric function that arises frequently is the absolute value function , written as \(|x|\). The absolute value function is defined as
\[f(x)=\begin{cases} -x, & \text{if }x<0 \\ x, & \text{if } x≥0 \end{cases} \nonumber \]
Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if \(x<0\), then \(|x|=−x>0,\) and if \(x>0\), then \(|x|=x>0.\) However, for \(x=0,\) \(|x|=0.\) Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if \(x=0\), the output \(|x|=0\). We conclude that the range of the absolute value function is \(\{y\,|\,y≥0\}.\) In Figure \(\PageIndex{13}\), we see that the absolute value function is symmetric about the \(y\)-axis and is therefore an even function.
Find the domain and range of the function \(f(x)=2|x−3|+4\).
Since the absolute value function is defined for all real numbers, the domain of this function is \((−∞,∞)\). Since \(|x−3|≥0\) for all \(x\), the function \(f(x)=2|x−3|+4≥4\). Therefore, the range is, at most, the set \(\{y\,|\,y≥4\}.\) To see that the range is, in fact, this whole set, we need to show that for \(y≥4\) there exists a real number \(x\) such that
\(2|x−3|+4=y\)
A real number \(x\) satisfies this equation as long as
\(|x−3|=\frac{1}{2}(y−4)\)
Since \(y≥4\), we know \(y−4≥0\), and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore,
\(|x−3|=\begin{cases} −(x−3), & \text{if } x<3\\x−3, & \text{if } x≥3\end{cases}\)
Therefore, we see there are two solutions:
\(x=±\frac{1}{2}(y−4)+3\).
The range of this function is \(\{y\,|\,y≥4\}.\)
For the function \(f(x)=|x+2|−4\), find the domain and range.
\(|x+2|≥0\) for all real numbers \(x\).
Domain = \((−∞,∞)\), range = \(\{y\,|\,y≥−4\}.\)
\((g∘f)(x)=g\big(f(x)\big)\)
\(f(x)=\begin{cases}−x, & \text{if } x<0\\x, & \text{if } x≥0\end{cases}\)
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A literature review section as part of a longer report should provide context and support a rationale for the new study. In a health sciences journal article, this section can sometimes be very short; in a dissertation, there is usually a whole chapter as a literature review, but prior literature should also be used throughout - for example to support methods and discussion sections.
A literature review is a critical summary and evaluation of the existing research (e.g., academic journal articles and books) on a specific topic. It is typically included as a separate section or chapter of a research paper or dissertation, serving as a contextual framework for a study. Literature reviews can vary in length depending on the ...
What kinds of literature reviews are written? Narrative review: The purpose of this type of review is to describe the current state of the research on a specific topic/research and to offer a critical analysis of the literature reviewed. Studies are grouped by research/theoretical categories, and themes and trends, strengths and weakness, and gaps are identified.
A literature review is a survey of scholarly sources (such as books, journal articles, and theses) related to a specific topic or research question. It is often written as part of a thesis, dissertation, or research paper, in order to situate your work in relation to existing knowledge.
Examples of literature reviews. Step 1 - Search for relevant literature. Step 2 - Evaluate and select sources. Step 3 - Identify themes, debates, and gaps. Step 4 - Outline your literature review's structure. Step 5 - Write your literature review.
A literature or narrative review is a comprehensive review and analysis of the published literature on a specific topic or research question. The literature that is reviewed contains: books, articles, academic articles, conference proceedings, association papers, and dissertations. It contains the most pertinent studies and points to important ...
A literature review is a review and synthesis of existing research on a topic or research question. A literature review is meant to analyze the scholarly literature, make connections across writings and identify strengths, weaknesses, trends, and missing conversations. A literature review should address different aspects of a topic as it ...
The purpose of a bibliometric review is to conduct an analysis of the academic literature in order to quantify and evaluate the scholarly output and influence of the literature. Bibliometric reviews aim to identify the most prolific and influential authors, research groups, and institutions within a particular field.
"A literature review is an account of what has been published on a topic by accredited scholars and researchers. In writing the literature review, your purpose is to convey to your reader what knowledge and ideas have been established on a topic, and what their strengths and weaknesses are. As a piece of writing, the literature review must be ...
A literature review is a written work that: Compiles significant research published on a topic by accredited scholars and researchers; Surveys scholarly articles, books, dissertations, conference proceedings, and other sources; Examines contrasting perspectives, theoretical approaches, methodologies, findings, results, conclusions.
Literature reviews are not created to produce new insights. They are written to explore and explain the literature on the topic or issue. One of the most important functions of a literature review is to lay the groundwork, provide background and context, for a larger research project such as a Masters thesis or PhD dissertation.
Most literature reviews are embedded in articles, books, and dissertations. In most research articles, there are set as a specific section, usually titled, "literature review", so they are hard to miss.But, sometimes, they are part of the narrative of the introduction of a book or article. This section is easily recognized since the author is engaging with other academics and experts by ...
The purpose of a literature review is to: Provide a foundation of knowledge on a topic; Identify areas of prior scholarship to prevent duplication and give credit to other researchers; Identify inconstancies: gaps in research, conflicts in previous studies, open questions left from other research;
The literature review chapter has a few important functions within your dissertation, thesis or research project. Let's take a look at these: Purpose #1 - Demonstrate your topic knowledge. The first function of the literature review chapter is, quite simply, to show the reader (or marker) that you know what you're talking about. In other ...
A literature review serves two main purposes: 1) To show awareness of the present state of knowledge in a particular field, including: seminal authors. the main empirical research. theoretical positions. controversies. breakthroughs as well as links to other related areas of knowledge. 2) To provide a foundation for the author's research.
Review and Reinforce. The goal of the literature review is to present an argument defending the relevance and value of a research question. To that end, a literature review must be balanced. For example, in proposing a new theory, both findings that are consistent with that theory and contradictory evidence must be discussed.
A literature review may consist of simply a summary of key sources, but in the social sciences, a literature review usually has an organizational pattern and combines both summary and synthesis, often within specific conceptual categories.A summary is a recap of the important information of the source, but a synthesis is a re-organization, or a reshuffling, of that information in a way that ...
A formal literature review is an evidence-based, in-depth analysis of a subject. There are many reasons for writing one and these will influence the length and style of your review, but in essence a literature review is a critical appraisal of the current collective knowledge on a subject. Rather than just being an exhaustive list of all that ...
Literature review purpose. The purpose of a literature review is to gain an understanding of the existing research and debates relevant to a particular topic or area of study, and to present that knowledge in the form of a written report. Conducting a literature review helps you build your knowledge in your field.
A literature review is a critical and comprehensive analysis of existing research, studies, articles, books, and other relevant sources on a specific topic or subject. It serves as a foundational step in the research process, helping researchers understand the current state of knowledge, identify gaps in the literature, and establish a context ...
A literature review is a critical analysis and synthesis of existing research on a particular topic. It provides an overview of the current state of knowledge, identifies gaps, and highlights key findings in the literature. 1 The purpose of a literature review is to situate your own research within the context of existing scholarship ...
Purpose and Importance of the Literature Review. An understanding of the current literature is critical for all phases of a research study. Lingard 9 recently invoked the "journal-as-conversation" metaphor as a way of understanding how one's research fits into the larger medical education conversation. As she described it: "Imagine yourself joining a conversation at a social event.
In addition to critical thinking, a literature review needs consistency, for example in the choice of passive vs. active voice and present vs. past tense. Rule 7: Find a Logical Structure. Like a well-baked cake, a good review has a number of telling features: it is worth the reader's time, timely, systematic, well written, focused, and ...
A literature review basically has three functions: to convey to the reader what knowledge and ideas have been established on a topic, and what their strengths and weaknesses are; in doing so, you clearly pass the message to the reader that you are familiar with these theories & ideas. Consequently, you are somewhat more of an expert writer, or ...
Background: Rheumatoid arthritis-associated interstitial lung disease (RA-ILD) is an important extra-articular manifestation of rheumatoid arthritis (RA). Identifying patients at risk of progression and death is crucial for improving RA-ILD management and outcomes. This paper explores current evidence on prognostic factors in RA-ILD. Methods: We conducted a systematic literature review to ...
Figure \(\PageIndex{2}\): A function maps every element in the domain to exactly one element in the range. Although each input can be sent to only one output, two different inputs can be sent to the same output.
Over 300 genes have been identified to be involved in the function and morphology of the cilia, and at least 40 genes are involved in the inappropriate function of the cilia . Dynein axonemal heavy chain 5 (DNAH5) and dynein axonemal intermediate chain 1 (DNAI1) are the most disease-causing genes involved in the pathogenesis of PCD .