Functions Toolkit (AQA GCSE 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