Composite & Inverse Functions | CIE IGCSE Maths: Extended Revision Notes 2023

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Composite Functions

What is a composite function?

A composite function is one function applied to the output of another function
Composite functions may also referred to as compound functions

What do composite functions look like?

The notation you will see for a composite function is fg(x)
- This can be written as f(g(x)) and means “f applied to the output of g(x)”
- i.e. g(x) happens first

Always apply the function on the outside to the output of the function on the inside
- gf(x) means g(f(x)) and means “g applied to the output of f(x)”
- i.e. f(x) happens first

How does a composite function work?

If you are putting a number into fg(x)
- STEP 1
  Put the number into g(x)
- STEP 2
  Put the output of g(x) into f(x)
- For example, if $f (x) = 2 x + 1$ and $g (x) = \frac{1}{x}$
  - $fg (2) = f (\frac{1}{2}) = 2 \times \frac{1}{2} + 1 = 2$
  - $gf (2) = g (2 \times 2 + 1) = g (5) = \frac{1}{5}$

If you are using algebra, to find an expression for a composite function
- STEP 1
  For fg(x) put g(x) wherever you see x in f(x)
- STEP 2
  Simplify if necessary
- For example, if $f (x) = 2 x + 1$ and $g (x) = \frac{1}{x}$
  - $fg (x) = f (\frac{1}{x}) = 2 \times \frac{1}{x} + 1 = \frac{2}{x} + 1$
  - $gf (x) = g (2 x + 1) = \frac{1}{2 x + 1}$

Examiner Tip

Make sure you are applying the functions in the correct order
- The letter nearest the bracket is the function applied first

Worked example

In this question, $f (x) = 2 x - 1$ and $g (x) = {(x + 2)}^{2}$ .

(a)

Find

fg (4)

g is on the inside of the composite function so apply g first.

fg (4) = f (g (4)) = f ({(4 + 2)}^{2}) = f (6^{2}) = f (36)

Apply f to the output of g.

\begin{array}{rcl} f (36) & = & 2 (36) - 1 \\ = & 72 - 1 \end{array}

fg (4) = 71

(b)

Find

g f (x)

f is on the inside of the composite function so apply f first by substituting the function f(x) into g(x).

g f (x) = g (f (x)) = g (2 x - 1) = {((2 x - 1) + 2)}^{2}

Simplify.

\begin{array}{rcl} g f (x) & = & {(2 x - 1 + 2)}^{2} \end{array}

gf (x) = (2 x + 1) 2

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Inverse Functions

What is an inverse function?

An inverse function does the exact opposite of the function it came from
- For example, if the function “doubles the number and adds 1” then its inverse is
- “subtract 1 and halve the result”
It is the inverse operations in the reverse order

How do I write inverse functions?

An inverse function f^-1 can be written as $f^{- 1} (x) = \dots$ or $f^{- 1} : x \mapsto \dots$
- For example, if $f (x) = 2 x + 1$ its inverse can be written as
- $f^{- 1} (x) = \frac{(x - 1)}{2}$ or $f^{- 1} : x \mapsto \frac{(x - 1)}{2}$

How do I find an inverse function?

The easiest way to find an inverse function is to 'cheat' and swap the $x$ and $y$ variables
- Note that this is a useful method but you MUST remember not to do this in any other circumstances in maths
- STEP 1
  Write the function in the form $y = \dots$
- STEP 2
  Swap the $x$ s and $y$ s to get $x = \dots$
- STEP 3
  Rearrange the expression to make $y$ the subject again
- STEP 4
  Write as f^-1(x) = … (or f^-1 : x ↦ …)
  - $y$ should not exist in the final answer
For example, if $f (x) = 2 x + 1$ its inverse can be found as follows
- STEP 1
  Write the function in the form $y = 2 x + 1$
- STEP 2
  Swap the $x$ and $y$ to get $x = 2 y + 1$
- STEP 3
  Rearrange the expression to make $y$ the subject again

$\begin{array}{rcl} x - 1 & = & 2 y \\ \frac{x - 1}{2} & = & y \to y = \frac{x - 1}{2} \end{array}$

- STEP 4
  Rewrite using the correct notation for an inverse function
  - $f^{- 1} (x) = \frac{x - 1}{2}$

How does a function relate to its inverse?

If $f (3) = 10$ then the input of 3 gives an output of 10
- The inverse function undoes f(x)
- An input of 10 into the inverse function gives an output of 3
  - If $f (3) = 10$ then $f^{- 1} (10) = 3$
${ff}^{- 1} (x) = f^{- 1} f (x) = x$
- If you apply a function to x, then immediately apply its inverse function, you get x
  - Whatever happened to x gets undone
- f and f^-1 cancel each other out when applied together
If $f (x) = 2^{x}$ and you want to solve $f^{- 1} (x) = 5$
- Finding the inverse function $f^{- 1} (x)$ in this case is tricky (impossible if you haven't studied logarithms)
- instead, take f of both sides and use that ${ff}^{- 1}$ cancel each other out:

$\begin{array}{rcl} {ff}^{- 1} (x) & = & f (5) \\ x & = & f (5) \\ x & = & 2^{5} = 32 \end{array}$

Worked example

Find the inverse of the function $f (x) = 5 - 3 x$ .

Write the function in the form $y = 5 - 3 x$ and then swap the $x$ and $y .$

$y = 5 - 3 x x = 5 - 3 y$

Rearrange the expression to make $y$ the subject again.

$\begin{array}{rcl} x & = & 5 - 3 y \\ x + 3 y & = & 5 \\ 3 y & = & 5 - x \\ y & = & \frac{5 - x}{3} \end{array}$

Rewrite using the correct notation for an inverse function

$f^{- 1} (x) = \frac{5 - x}{3}$

Composite & Inverse Functions (CIE IGCSE Maths: Extended)

Revision Note

Composite Functions

What is a composite function?

What do composite functions look like?

How does a composite function work?

Examiner Tip

Worked example

Inverse Functions

What is an inverse function?

How do I write inverse functions?

How do I find an inverse function?

How does a function relate to its inverse?

Worked example

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Author: Amber