[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207516-en":3,"doc-seo-207516-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207516,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",4,"Exam","AQA GCSE Further Maths - Functions Toolkit - Introduction to Functions","Functions Toolkit provides focused revision notes for AQA GCSE Further Maths on the topic of functions. It explains what a function is using the “machine” idea, distinguishes inputs and outputs, and shows common notations such as f(x), g(x) and y = … . The material then demonstrates how to evaluate functions, form equations to find inputs, and represent functions with graphs. It also introduces domain and range, including how to describe them using inequalities and how to deduce range from an expression and an allowed domain, supported by worked examples and examiner tips.","Head to www.savemyexams. com for more awesome resources  \n AQA GCSE Further Maths  \nFunctions  \nContents  \n Functions Toolkit  \n Composite & Inverse Functions  \nPage 1 of 20  \n© 2015 − 2025 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  \nYour notes  \nHead to www.savemyexams. com for more awesome resources  \nFunctions Toolkit  \nIntroduction to Functions What isa function?  \n A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers  \n It may be thought of as a mathematical “machine”  \n For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are \" multiply by 2 (×2) \" and \" add 1 (+1) \"  \n Putting 3 into the function would give 2 × 3 + 1 = 7  \n Putting-4 in would give 2 × (-4) + 1 = -7  \n Putting x in would give2x + 1  \n The number being put into the function is often called the input  \n The number coming out of the function is often called the output What does a function look like?  \n A function f can be written asf(x) = …  \n Other letters can be used. g, h and j are common but any letter can technically be used  \n Normally, a new letter will be used to de|ne a new function in a question  \n For example, the function with the rule “triple the number and subtract 4” would be written  \n f( x) = 3x – 4  \n In such cases, x would bethe input and f( x) would bethe output  \n Sometimes functions don’t have names like f and are just written as y = …  \n eg. y = 3x – 4 How does a function work?  \n A function hasan input ( x) and output (f( x) or y)  \n Whatever goes in the bracket (instead of x)with f, replaces the x on the other side  \nPage 2 of 20  \n© 2015 − 2025 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  \nYour notes  \nHead to www.savemyexams. com for more awesome resources  \nThis is the input  \nIf the input is known, the output can be calculated  \n For example, given the function f( x) = 2x + 1  \n f(3) = 2 × 3 + 1 = 7  \n f( − 4) = 2 × ( − 4) + 1 = − 7  \n f( a) = 2a + 1  \nIf the output is known, an equation can be formed and solved to |nd the input  \n For example, given the function f( x) = 2x + 1  \n If f( x) = 15 , the equation2x + 1 = 15 can be formed  \n Solving this equation gives an input of 7  \nWorked Example  \nA function is de|ned asf( x) = 3x2 − 2x + 1.  \n(a) Find f(7) .  \nThe input is x = 7 , so substitute 7 into the expression everywhere you see an x .  \nf(7) = 3 (7)2 − 2 (7) + 1  \nCalculate.  \nf(7) = 3 (49) − 14 + 1 = 147 − 14 + 1  \nf (7) = 134  \n(b) Find f( x + 3) .  \nThe input is x = x + 3 so substitute x + 3 into the expression everywhere you see an x .  \nf( x + 3) = 3 ( x + 3)2 − 2 ( x + 3) + 1  \nExpand the brackets and simplify.  \nPage 3 of 20  \n© 2015 − 2025 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  \nYour notes  \nHead to www.savemyexams. com for more awesome resources  \nf( x + 3) = 3 ( x2 + 6x + 9) − 2 ( x + 3) + 1 = 3x2 + 18x + 27 − 2x − 6 + 1  \n= 3x2 + 16x + 22  \nf (x + 3) = 3x2 + 16x + 22  \nA second function is de|ned g ( x) = 3x – 4.  \n(c) Find the value of x for which g ( x) = − 16.  \nForm an equation by setting the function equal to-16 .  \n3x − 4 = − 16  \nSolve the equation by |rst adding 4 to both sides, then dividing by 3 .  \n3x − 4 = − 16 3x = − 12  12  \nx = − 3  \nx = − 4  \nDomain & Range  \nHow are functions related to graphs?  \n Functions can be represented as graphs on x andy axes  \n The x-axis values are the inputs  \n They-axis values are the outputs  \n To see what graph to plot, replace f(x) = . .. with y = . . .  \nPage 4 of 20  \n© 2015 − 2025 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  \nYour notes  \nHead to www.savemyexams. com for more awesome resources  \nWhat is the domain of a function?  \n The domain of a function is theset of all inputs that the function is allowed to take  \n Domains can be described in words  \n they must refer to x  \n you can use inequality signs if needed  \n you can ","cbCaivbZI0Io4DdZ","https://ap.wps.com/l/cbCaivbZI0Io4DdZ","pdf",1205065,1,20,"English","en",105,"# Functions Toolkit\n## Introduction to Functions\n## Composite & Inverse Functions\n## Domain & Range\n## Examiner Tips and Tricks","[{\"question\":\"What is a function in GCSE Further Maths?\",\"answer\":\"A function is a combination of mathematical operations that takes an input set of numbers and produces an output set. It can be viewed as a machine that transforms inputs into outputs.\"},{\"question\":\"How do you evaluate a function like f(x) = 2x + 1?\",\"answer\":\"Substitute the given input value for x everywhere in the expression. For example, f(3) = 2×3 + 1 = 7.\"},{\"question\":\"How do you find the domain and range of a function?\",\"answer\":\"The domain is the set of allowed inputs (x-values) and is described using statements or inequalities. The range is the set of possible outputs (f(x) values) and is deduced based on the domain, often by sketching the graph or using a table of values.\"}]","AQA GCSE Further Maths - Functions Toolkit - Introduction to Functions | PDF",1788599820,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"aqa-gcse-further-maths-functions-toolkit-introduction-to-functions","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/aqa-gcse-further-maths-functions-toolkit-introduction-to-functions/207516/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is a function in GCSE Further Maths?","Question",{"text":74,"@type":75},"A function is a combination of mathematical operations that takes an input set of numbers and produces an output set. It can be viewed as a machine that transforms inputs into outputs.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do you evaluate a function like f(x) = 2x + 1?",{"text":79,"@type":75},"Substitute the given input value for x everywhere in the expression. For example, f(3) = 2×3 + 1 = 7.",{"name":81,"@type":72,"acceptedAnswer":82},"How do you find the domain and range of a function?",{"text":83,"@type":75},"The domain is the set of allowed inputs (x-values) and is described using statements or inequalities. 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