Functions Toolkit | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Introduction to Functions

What is a function?

• A function represents a set of mathematical operations that transform one set of numbers into another set.
  • It can be visualized as a mathematical "machine" where specific rules are applied to input numbers to produce output numbers.
  • For instance, consider a function with the rule "double the number and add 1." Here, the operations involved are "multiply by 2" and "add 1."
  • When 3 is input into this function: 2 x 3 + 1 = 7
  • When -4 is input: 2 x (-4) + 1 = -7
• The number input into the function is commonly termed the input, while the resulting number is referred to as 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
  • f(x) = 3x − 4 or f: x ↦ 3x − 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 = 3x − 4.

How does a function work?

• A function has an input (x) and an output (f(x) or y). 
• Whatever is inside the bracket with f replaces x in the function. 
• This value is the input. 
• If the input is known, the output can be calculated. 
• For example, given f(x) = 2x + 1: 
  • f(3) = 2 ⋅ 3 + 1 = 7 
  • f(−4) = 2 ⋅ (−4) + 1 = −7 
  • f(a) = 2a + 1 
• If the output is known, an equation can be formed to find the input. 
• For example, given f(x) = 2x + 1: 
  • If f(x) = 15, then the equation 2x + 1 = 15 can be formed. 
  • Solving this equation gives an input of 7.

What is a mapping diagram?

• A mapping diagram illustrates how an 'input' from one set corresponds to an 'output' in another set.
• For example, a mapping diagram for the function x + 3 where x ≥ 3 could be shown as:

Domain & Range

• Functions are graphically represented on x and y axes.
  • The x-axis values serve as the inputs.
  • The y-axis values represent the outputs.
• To plot the graph, substitute f(x) with y =

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 x 
    • 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/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) = √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)
•  must exclude x = 1 and x = -7 from any domain
  • These two inputs make the function undefined (dividing by zero)
•  must exclude x < 3 from any domain
  • Any input in x < 3 leads to square-rooting a negative

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

• 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