Chapter 21: Functions

Functions

• You need to be able to work with function notation confidently in your exam

Composite functions

• If you apply two functions one after the other, you can write a single function which has the same effect as the two combined functions
• This is called a composite function

Working it out

• Order is important in composite functions
• You can think of fg(x) as f(g(x))
• You work out g(x) first then you use this answer as your input for f(x)
• To find an algebraic expression for fg(x) you need to substitute the whole expression for g(x) for each instance of x in the expression for f(x)

Inverse functions

• For a function f the inverse of f is the function that undoes f
• You write the inverse of f(-1)
• If you apply f then f(-1) you will end up back where you started
  • If you apply f then f(-1) you have applied the composite function f(-1)f
  • The output of f(-1)f is the same as the input

Finding the inverse

• To find the inverse of a function given in the form f(x) you need to
  • Write the function in the form y=
  • Rearrange to make x the subject
  • Swap any y’s for x’s and rewrite as f(-1)(x) =

Methods

• You can sometimes use a flow chart to find an inverse
• Here is an example flow chart for f(x)
• +3, x5
• You would then work backwards through the flow chart to find f(-1)(x)
• /5 -3
• You subtract 3 then divide by 5
  • You could then write this in algebraic form

\