[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573097-105":53,"doc-detail-573097-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","functions-and-graphs-cheat-sheet-chapter-2-study-notes","Functions and Graphs Cheat Sheet - Chapter 2 - Study Notes","","Functions and Graphs Cheat Sheet explains what a function is as a type of mapping, distinguishing one-to-one, many-to-one, and rejecting one-to-many as non-functions. It defines domain and range and shows common function notations, then covers composition of functions and why order matters. The sheet also introduces modulus operators and modulus functions with sketching algorithms, followed by inverse functions, including key properties and a step-by-step method to find inverses and ranges. Examples support each concept.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/functions-and-graphs-cheat-sheet-chapter-2-study-notes/573097/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/functions-and-graphs-cheat-sheet-chapter-2-study-notes/573097.png","ImageObject",442,249,{"name":88,"@type":89},"Angel","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do mappings relate to functions?","Question",{"text":108,"@type":109},"Functions are mappings that are one-to-one or many-to-one. One-to-many mappings are not functions because a single input would correspond to multiple outputs.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What are domain and range of a function?",{"text":113,"@type":109},"The domain is the set of all possible inputs (the x-values the function can take). The range is the set of all possible outputs (the y-values the function can produce).",{"name":115,"@type":106,"acceptedAnswer":116},"How do you compose two functions and why does order matter?",{"text":117,"@type":109},"To form a composite function, apply one function first and then the other. In general, f∘g is not equal to g∘f, because the order of application changes the result.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573097,1792053274,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Functions and Graphs Cheat Sheet  \nWhat really is a function?  \nBefore we can properly define a function, we must first understand mappings.  \nA mapping takes a set of numbers and transforms them into another set of numbers. Mappings are often represented by algebraic equations. For example, 􀝕 = 􀝔 2 is a mapping that takes every real number and maps them to their respective squares.  \nThere are three different types of mappings you need to be well-acquainted with:  \n\n| Mapping type Description |  | One-to-one\u003Cbr>Each input has a unique output. | Many-to-one\u003Cbr>Each input has just one associated output, but there are some inputs that are mapped to the same output. | One-to-many\u003Cbr>There exist inputs that are mapped to more than one output. |\n| --- | --- | --- | --- | --- |\n| Example | \u003Cbr>􀝕 = 􀝔 |  | | |\n\n• Functions are mappings that are one-to-one or many-to-one. One-to-many mappings are not functions!  \nFunction notation  \nYou need to understand what is meant by the domain and range of a function:  \n▪ The domain of a function is the set of all possible inputs (i. e. the set of x-values the function takes)  \n▪ The range of a function is the set of all possible outputs (i. e. the set of y-values the function takes) There are two main types of notation you will see for denoting functions:  \n▪ 􀝂 ( 􀝔) = √􀝔,  {􀢞 ∈ ℝ, 􀢞 ≥ 􀫙}  \n▪ 􀝂: 􀝔 ⟼ √􀝔 , {􀢞 ∈ ℝ, 􀢞 ≥ 􀫙}  \nThese notations are equivalent for one-to-one or many-to-one functions. The “ ⟼ \" symbol means that each element in the domain is mapped to just one element in the range. The bold text represents the domain of the function.  \nComposition of functions  \nWe can combine two or more functions to make a new function. This new function is called a composite function.  \nLet’s take two functions, 􀝃(􀝔) and 􀝂(􀝔) . If we wish to combine them, there are two ways we can do this:  \n▪ 􀝂􀝃 (􀝔) we interpret this as applying 􀝃 (􀝔) first, then 􀝂 (􀝔).  \n▪ 􀝃􀝂 (􀝔) we interpret this as applying 􀝂(􀝔) first, then 􀝃 (􀝔) .  \nIt is important to realise that 􀝃􀝂( 􀝔) ≠ 􀝂􀝃 ( 􀝔) in general.  \nWhen composing functions, it is helpful to interpret 􀝂􀝃 ( 􀝔) as 􀝂[􀝃 ( 􀝔) ]. This means that we must replace the inputs of 􀝂(􀝔) with 􀝃 (􀝔), to get 􀝂􀝃 ( 􀝔) .  \nBelow is an example of how we do this in practice.  \nExample 1: Given that 􀝂( 􀝔) = 3􀝔 − 2 and 􀝃( 􀝔) = 􀝔 2 + 4􀝔 − 2, find an expression for:  \n(i) 􀝂􀝃 ( 􀝔)  \n(ii) 􀝃􀝂 (􀝔)  \n(i) 􀝂􀝃 (􀝔) is found by replacing all the 􀝔 terms in 􀝂􀝃(􀝔) = 􀝂[􀝃(􀝔)] = 􀝂(􀝔 2 + 4􀝔 − 2)􀝂 (􀝔) with 􀝔 2 + 4􀝔 − 2  \nDoing so: 􀝂(􀝔2 + 4􀝔 − 2) = 3(􀝔2 + 4􀝔 − 2) − 2  \nSimplifying gives: 􀝂(􀝔2 + 4􀝔 − 2) = 3􀝔2 + 12􀝔 − 6 − 2  \nThe final answer is: so 􀝂􀝃(􀝔) = 3􀝔2 + 12􀝔 − 8  \n(ii) 􀝃􀝂(􀝔) is found by replacing all the 􀝔 terms in 􀝃􀝂 ( 􀝔) = 􀝃 [􀝂 ( 􀝔)] = 􀝃 ( 3􀝔 − 2)  \n􀝃 (􀝔) with 3􀝔 − 2  \nDoing so: 􀝃 (3􀝔 − 2) = (3􀝔 − 2)2 + 4(3􀝔 − 2) − 2  \nSimplifying gives: 􀝃 (3􀝔 − 2) = 9􀝔2 − 12􀝔 + 4 + 12􀝔 − 8 − 2 = 9􀝔2 − 6  \nThe final answer is: 􀝏􀝋 􀝃􀝂(􀝔) = 9􀝔2 − 6  \nIt is clear from this example that 􀝃􀝂(􀝔) ≠ 􀝂􀝃 (􀝔) .  \nThe modulus operator  \nThe modulus of any number k, denoted as |k|, is simply its non-negative value. For example, |−3| = 3 and also |3| = 3. The modulus is sometimes called the absolute value.  \nModulus functions  \nThere are two types of modulus functions you will encounter. The first is functions of the form 􀝕 = |􀝂( 􀝔)|, and the second is functions of the form 􀝕 = 􀝂 (|􀝔|) . Let’s discuss each of these, starting with the former.  \n􀝕 = |􀝂(􀝔) | e.g 􀝕 = |5􀝔 − 4|  \nWith functions of this form, we can see that the modulus is placed around the entire function. This means that the function will never output a negative number. Graphically, this means that the function will always be above (or on) the x-axis.  \n􀝕 = 􀝂 (|􀝔|) e.g 􀝕 = sin |􀝔|  \nHere, the modulus is applied to the inputs of the function, rather than the outputs. This means that for any number a, we have that 􀝂( 􀜽) =􀝂 ( −􀜽) . Graphically, this means that the function is symmetric about the y-axis.  \nThe above interpretations lead us onto the following algorithms for sket","cbCaicPdJoLupMDm","https://ap.wps.com/l/cbCaicPdJoLupMDm","pdf",1092176,"English","# What really is a function?\n## Mappings and function types\n# Function notation and domain/range\n# Composition of functions\n# The modulus operator and modulus functions\n## Sketching algorithms\n# Inverse functions\n## Finding an inverse","[{\"question\":\"How do mappings relate to functions?\",\"answer\":\"Functions are mappings that are one-to-one or many-to-one. One-to-many mappings are not functions because a single input would correspond to multiple outputs.\"},{\"question\":\"What are domain and range of a function?\",\"answer\":\"The domain is the set of all possible inputs (the x-values the function can take). The range is the set of all possible outputs (the y-values the function can produce).\"},{\"question\":\"How do you compose two functions and why does order matter?\",\"answer\":\"To form a composite function, apply one function first and then the other. In general, f∘g is not equal to g∘f, because the order of application changes the result.\"}]","Functions and Graphs Cheat Sheet - Chapter 2 - Study Notes | PDF",1791932207]