Functions Toolkit (Cambridge O Level Maths)

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 in would give
• 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
  •    or   
• In such cases,  would be the input and  would be the output
• Sometimes functions don’t have names like f and are just written as y = …
  • eg. 

How does a function work?

• A function has an input and output 
• Whatever goes in the bracket (instead of )with f, replaces the on the other side
  • This is the input
• If the input is known, the output can be calculated
  • For example, given the function 
    • 
    • 
    • 

• If the output is known, an equation can be formed and solved to find the input
  • For example, given the function 
    
    • If , the equation can be formed
    • Solving this equation gives an input of 7

What is a mapping diagram?

• A mapping diagram takes an ‘input’ from one set of values to an ‘output’ in another
• For example, a mapping diagram for the function  where could be shown as:

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 =

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) =  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) =  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) =  must exclude x = 1 and x = -7 from any domain
    • These two inputs make the function undefined (dividing by zero)
  • f(x) =  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