GCSEMathsAQARevision Notes2. AlgebraFunctionsFunctions Toolkit

Functions Toolkit (AQA GCSE Maths)

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Introduction to Functions

What is a function?

  • 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
    • It may be thought of as a mathematical “machine”
    • 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)"  
      • Putting 3 in to the function would give 2 × 3 + 1 = 7
      • Putting -4 in would give 2 × (-4) + 1 = -7 
    • Putting x in would give 2 x space plus space 1
  • The number being put into the function is often called the input
  • The number coming out of the function is often called the output

What does a function look like?

  • A function f can be written as f(x) = … or f : x ↦ …
    • These two different types of notation mean exactly the same thing
    • Other letters can be used. g, h and j are common but any letter can technically be used
      • Normally, a new letter will be used to define a new function in a question
  • For example, the function with the rule “triple the number and subtract 4” would be written
    • straight f left parenthesis x right parenthesis space equals space 3 x space – space 4   or   straight f colon x space rightwards arrow from bar space 3 x space – space 4
  • In such cases, x would be the input and straight f open parentheses x close parentheses would be the output
  • Sometimes functions don’t have names like f and are just written as y = …
    • eg. y space equals space 3 x space – space 4

How does a function work?

  • A function has an input open parentheses x close parentheses and output left parenthesis straight f open parentheses x close parentheses space or space space y right parenthesis
  • Whatever goes in the bracket (instead of x)with f, replaces the x on the other side
    • This is the input
  • If the input is known, the output can be calculated
    • For example, given the function straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1
      • straight f left parenthesis 3 right parenthesis space equals space 2 space cross times space 3 space plus space 1 equals 7
      • straight f left parenthesis negative 4 right parenthesis space equals space 2 space cross times space left parenthesis negative 4 right parenthesis space plus space 1 space equals space minus 7
      • straight f left parenthesis a right parenthesis space equals space 2 a space plus space 1
  • If the output is known, an equation can be formed and solved to find the input
    • For example, given the function straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1
      • If straight f left parenthesis x right parenthesis space equals space 15, the equation 2 x space plus space 1 space equals space 15 can be formed
      • Solving this equation gives an input of 7

Worked example

A function is defined as straight f open parentheses x close parentheses space equals space 3 x to the power of 2 space end exponent minus space 2 x space plus space 1.

(a)
Find straight f open parentheses 7 close parentheses.
  
The input is x space equals space 7, so substitute 7 into the expression everywhere you see an x.
  
straight f open parentheses 7 close parentheses space equals space 3 open parentheses 7 close parentheses squared space minus space 2 open parentheses 7 close parentheses space plus space 1
  
Calculate.
  
table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses 7 close parentheses space end cell equals cell space 3 open parentheses 49 close parentheses space minus space 14 space plus space 1 end cell row blank equals cell space 147 space minus space 14 space plus space 1 end cell end table
  
bold f begin bold style stretchy left parenthesis 7 stretchy right parenthesis end style bold space bold equals bold space bold 134

(b)
Find straight f open parentheses x space plus space 3 close parentheses.
   
The input is x space equals space x space plus space 3 so substitute space x space plus space 3 into the expression everywhere you see an x.
 

straight f open parentheses x space plus space 3 close parentheses space equals space 3 open parentheses x space plus space 3 close parentheses squared space minus space 2 open parentheses x space plus space 3 close parentheses space plus space 1
 

Expand the brackets and simplify.
 

table row cell straight f open parentheses x space plus space 3 close parentheses space end cell equals cell space 3 open parentheses x squared space plus space 6 x space plus space 9 close parentheses space minus space 2 open parentheses x space plus space 3 close parentheses space plus space 1 end cell row blank equals cell space 3 x squared space plus space 18 x space plus space 27 space minus space 2 x space minus space 6 space plus space 1 end cell row blank equals cell space 3 x squared space plus space 16 x space plus space 22 end cell end table
 

A second function is defined straight g space colon space x space rightwards arrow from bar space 3 x space – space 4.

(c)
Find the value of x for which straight g space colon space x space rightwards arrow from bar space minus 16.
  
Form an equation by setting the function equal to -16.
 
table row cell 3 x space minus space 4 space end cell equals cell space minus 16 end cell end table
 
Solve the equation by first adding 4 to both sides, then dividing by 3. 
 
table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x space minus space 4 space end cell equals cell space minus 16 end cell row cell 3 x space end cell equals cell space minus 12 end cell row cell x space end cell equals cell space minus 12 over 3 end cell end table
 
bold italic x bold space bold equals bold space bold minus bold 4

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    Daniel I

    Author: Daniel I

    Expertise: Maths

    Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.