Functions Toolkit (Edexcel IGCSE 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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Domain & Range

How are functions related to graphs?

  • Functions can be represented as graphs on x and y axes
    • The x-axis values are the inputs
    • The y-axis values are the outputs
  • To see what graph to plot, replace f(x) = with y =

DRE Notes fig1, downloadable IGCSE & GCSE Maths revision notes

What is the domain of a function?

  • The domain of a function is the set of all inputs that the function is allowed to take
  • Domains can be described in words
    • they must refer to
      • not y or f(x)
    • you can use "not equal to" ≠ if needed
    • you can use inequality signs if needed
  • Examples of domains are below:
    • f(x) = 3x + 2 takes any x value
      • the domain is "all values of x"
    • f(x) = 1 over x takes any x value except 0 (you cannot divide by 0)
      • the domain is "all values of x except 0", or simply "x ≠ 0"
    • f(x) = square root of x takes any x value that are not negative (you cannot square root a negative)
      • the domain is "x ≥ 0"
    • f(x) = x2 takes any x value (negative x values are fine as inputs)
      • the domain is "all values of x"
  • Some domains are restricted by choice
    • f(x) = 3x + 2 with the domain 0 < x < 5
      • This question wants to concentrate on that domain only (even though bigger domains exist)
  • Some domains must exclude certain values (or sets of values)
    • f(x) = fraction numerator 1 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 7 close parentheses end fraction must exclude x = 1 and x = -7 from any domain
      • These two inputs make the function undefined (dividing by zero)
    • f(x) = square root of x minus 3 end root must exclude x < 3 from any domain
      • Any input in x < 3 leads to square-rooting a negative

 DRE Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What is the range of a function?

  • The range of a function is the set of all outputs that the function gives out
  • Ranges can be described in words
    • they must refer to f(x) 
      • not x or y
    • you can use "not equal to" ≠ if needed
    • you can use inequality signs if needed
  • Ranges are influenced by domains
  • Examples of ranges are below:
    • f(x) = 3x + 2 with domain x > 0
      • The range is "f(x) > 2"
      • This is because if the inputs are all greater than 0, the outputs will all be greater than 2
      • This could be seen from a sketch or by substituting inputs of x > 0 into f(x)
    • f(x) = x2 with domain "all values of x"
      • The range is f(x) ≥ 0
      • This is because all values of x get squared (so no negative outputs are created)

How do I solve problems involving the domain and range?

  • You need to be able to identify and explain any exclusions in the domain of a function
  • You need to be able to deduce the range of a function from its expression and domain

DRE Notes fig4, downloadable IGCSE & GCSE Maths revision notes

  • You may also be asked to sketch a graph of a function
    • This could involve sketching parts of familiar graphs that are restricted because of the domain and exclusions

    DRE Notes fig5, downloadable IGCSE & GCSE Maths revision notes

Examiner Tip

  • A graph of the function can help “see” both the domain and range of function (a sketch can help if you have not been given a diagram)

Worked example

Two functions are given by
 

straight f open parentheses x close parentheses equals 10 minus x space space space space space space space space space space space space space space straight g open parentheses x close parentheses equals fraction numerator 1 over denominator 2 x minus 1 end fraction

(a)
If the domain of function f is 2 less than x less or equal than 4, find the range.
 
The domain is the set of inputs
Substitute x = 2 into f(x) to find its output
 
table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses 2 close parentheses end cell equals cell 10 minus 2 end cell row blank equals 8 end table
 
Substitute x = 4 into f(x) to find its output
 
table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses 4 close parentheses end cell equals cell 10 minus 4 end cell row blank equals 6 end table
 
Think of f(x) = 10 - x as a graph
 
the graph of y equals 10 minus x
 
This straight-line graph has a negative gradient
Between x = 2 and x = 4 the graph decreases from a height of 8 to a height of 6 
Relate this to outputs
 
all outputs are between 6 and 8
 
Write down the range using f(x)
Remember that the inequality is "equal to" at x = 4, f(x) = 6
the range is 
 
(b)
Write down the value of x that must be excluded from the domain of function g.
 
An input cannot cause the function to divide by zero
Find out when "dividing by zero" would happen
 
2 x minus 1 equals 0
 
Solve to find this value of x (the one that must be excluded)
 
2 x equals 1
x equals 1 half
bold italic x bold equals bold 1 over bold 2 must be excluded

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.