Composite & Inverse Functions | OCR GCSE Maths Revision Notes 2022

Composite Functions

What is a composite function?

Questions can involve more than one function
A composite function is one function applied to the output of another function
- The input goes through the 1st function to become an output
- This output goes through the 2nd function to become a new output
Composite functions may also be referred to as compound functions

Worked example

Two functions, A and B, are given by:

$Function A : input \to \times 3 \to - 8 \to output Function B : input \to + 5 \to \times 2 \to output$

A composite function, C, is given by

$Function C : input \to Function A \to Function B \to output$

(a)

If the input to function C is 7, find the output.

Substitute the number 7 into function A

7 × 3 - 8

Work out this value

Substitute 13 into function B

(13 + 5) × 2

Work out this value

the output of function C is 36

(b)

If the input to function C is x and the ouput is 4x, find the value of x.

Substitute the letter x into function A

x × 3 - 8

Simplify this expression

3x - 8

Substitute 3x - 8 into function B

(3x - 8 + 5) × 2

Simplify this expression

(3x - 3) × 2
= 6x - 6

The output is 6x - 6
The question says the output of function C is meant to be 4x
Set these two outputs equal to each other

6x - 6 = 4x

Solve this equation to find x

2x = 6
x = 3

x = 3

Inverse Functions

What is an inverse function?

An inverse function reverses (undoes) the operations of a function
- Let's say a function “doubles the input, then adds 1”
  - Its inverse function must "subtract 1, then halve the result"
The inverse of $input \to \times 6 \to + 4 \to output$ is $input \to - 4 \to \div 6 \to output$
- The order of the boxes is reversed
- The operation is replaced by its inverse operation
  - × ↔ ÷ and + ↔ -
If a number goes through a function, then that result goes through the inverse function, you get back the same number again!

How can I use algebra to find inverse functions?

If putting x into a function gives out y, then putting y into the inverse function gives back x
Inverse functions are related to rearranging formulae

- Let's say a function is $x \to \times 2 \to + 3 \to y$
  - It's inverse is therefore $x \to - 3 \to \div 2 \to y$
- In algebra, the original function is $2 x + 3 = y$
- Look what happens when you make x the subject
  - $x = \frac{y - 3}{2}$
  - This is the function $y \to - 3 \to \div 2 \to x$
  - The operations are identical to that of the inverse function! (the input here is a y though)
- Swapping the x and y shows the inverse function more clearly
  - $x \to - 3 \to \div 2 \to y$

Worked example

A function is given by

$input \to \times 4 \to - 5 \to output$

(a)

If the output is 7, find the input.

Method 1
Reverse the order and operations to find the inverse function

$input \to + 5 \to \div 4 \to output$

Substitute 7 as the input into the inverse function

(7 + 5) ÷ 4

Work out this value

Method 2
If the output is 7, let the input to the function be x

$\begin{array}{rcl} x \times 4 - 5 & = & 7 \\ 4 x - 5 & = & 7 \end{array}$

Solve this equation to find x

$\begin{array}{rcl} 4 x & = & 12 \\ x & = & 3 \end{array}$

(b)

Find an algebraic expression for the inverse function, where the input is x.

Method 1
Reverse the order and operations to find the inverse function

$input \to + 5 \to \div 4 \to output$

Use x as the input

(x + 5) ÷ 4

Work out this value

the inverse function is $\frac{x + 5}{4}$

Method 2
Write the original function in algebra (with x as the input and y as the output)

$\begin{array}{rcl} x \times 4 - 5 & = & y \\ 4 x - 5 & = & y \end{array}$

Make x the subject of this equation

$\begin{array}{rcl} 4 x & = & y + 5 \\ x & = & \frac{y + 5}{4} \end{array}$

This shows the "structure" of the inverse function, but currently uses a y
Replace the y with an x (the question wanted x as the input, not y)

the inverse function is $\frac{x + 5}{4}$

Composite & Inverse Functions (OCR GCSE Maths)

Revision Note

Composite Functions

What is a composite function?

Worked example

Inverse Functions

What is an inverse function?

How can I use algebra to find inverse functions?

Worked example

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Author: Mark