GCSE Maths Algebra Functions

Inverse Functions

# Inverse Functions

Here we will learn about inverse functions, including what an inverse function is, the notation used for an inverse function and how to find an inverse function.

There are also inverse function worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are inverse functions?

Inverse functions are functions which will give the input of a function when we know the output; it “undoes” the function.

We have seen how we use a function machine to work backwards to find the input from a known output. If we were to write the algebraic expression relating to these inverse operations in the correct order, we would have found the inverse function.

To write the inverse of the function f , we use the notation f^{-1} .

You may have already seen this notation used when doing trigonometry.

E.g.

When finding a missing angle or solving the equation sin (x)=0.6 , we would need to use the inverse of the sine function, x=sin^{-1}(0.6).

In GCSE mathematics, we will need to be able to find the inverse of a function such as f(x)=4x-2 and g(x)=\frac{x-2}{x+1} .

To find an inverse function we will use the method of rearranging formulae appropriate to the given function.

Inverse functions can be used to solve equations or find missing x values on graphs if we know the y value.

In A level mathematics instead of using sin^{-1}, cos^{-1} and tan^{-1}, the inverse trigonometric functions have different names. We would use arcsin, arccos and arctan.

We also look at when inverse functions can exist – functions such as quadratics, cubics, sin, cos and tan must have a restricted domain (a limit to which x values are allowed).

We also look at how to find the graph of the inverse function f^{-1} by reflecting the graph of the function f in the line y=x .

### What are inverse functions? ## How to find inverse functions

In order to find an inverse function:

1. Write out the expression for the original function using a y instead of the x . Set this expression equal to x.
2. Rearrange the equation to make y the subject.
3. Write your inverse function using the f^{-1} notation.

### Explain how to find inverse functions ### Related functions lessons

Inverse functions is part of the larger topic, functions. You may find it useful to explore the main topic before looking into the detailed individual lessons below:

## Inverse functions examples

### Example 1: Finding the inverse of a linear function

If f(x)=3x-7 , find f^{-1}(x).

1. Write out the expression for the original function using a y instead of the x . Set this expression equal to x.

3y-7=x

2Rearrange the equation to make y the subject.

\begin{aligned} & 3y-7=x \\\\ & 3y=x+7 \hspace{2cm} \text{Add } 7 \text{ to both sides of the equation} \\\\ & y=\frac{x+7}{3} \hspace{2cm} \text{Divide both sides by } 3 \end{aligned}

3Write your inverse function using the f^{-1} notation.

{{f}^{-1}}\left( x \right)=\frac{x+7}{3}

### Example 2: Finding the inverse of a linear function

If g(x)=9-\frac{x}{2} find g^{-1}(x) .

9-\frac{y}{2}=x

\begin{aligned} & 9-\frac{y}{2}=x \\\\ & 9-x=\frac{y}{2} \hspace{2cm} \text{Add } \frac{y}{2} \text{ to both sides and subtract } x \text{ from both sides. } \\\\ & 18-2x=y \hspace{1.8cm} \text{Multiply both sides by } 2. \end{aligned}

{{g}^{-1}}\left( x \right)=18-2x Use g^{-1} because the original function was g.

### Example 3: Finding the inverse of a function involving a power

If f(x)=2x^2-5 find f^{-1}(x) .

2{{y}^{2}}-5=x

\begin{aligned} & 2{{y}^{2}}-5=x \\\\ & 2{{y}^{2}}=x+5 \hspace{3cm} \text{Add } 5 \text{ to both sides.. } \\\\ & {{y}^{2}}=\frac{x+5}{2} \hspace{3cm} \text{Divide both sides by } 2. \\\\ & y=\sqrt{\frac{x+5}{2}} \hspace{2.8cm} \text{Square root both sides.} \end{aligned}

An inverse function cannot have two different outputs for one input so we only need the positive square root.

{{f}^{-1}}\left( x \right)=\sqrt{\frac{x+5}{2}}

### Example 4:  Finding the inverse of a function involving an algebraic fraction

If h(x)=\frac{x-3}{x+2} , find h^{-1}(x) .

\frac{y-3}{y+2}=x

\begin{aligned} & \frac{y-3}{y+2}=x \\\\ & y-3=x\left( y+2 \right) \hspace{2.7cm} \text{Multiply both sides by the denominator } y+2 \\\\ & y-3=xy+2x \hspace{2.8cm} \text{Expand the bracket} \\\\ & y-xy=2x+3 \hspace{2.8cm} \text{Isolate the } y \text{ by subtracting } xy \text{ from both sides}\\ & \hspace{5cm} \text{and adding } 3 \text{ to both sides} \\\\ & y\left( 1-x \right)=2x+3 \hspace{2.5cm} \text{Factor out the } y\\\\ & y=\frac{2x+3}{1-x} \hspace{3.4cm} \text{Divide both sides by } 1-x \end{aligned}

{{h}^{-1}}\left( x \right)=\frac{2x+3}{1-x}

### Common misconceptions

• The inverse function notation means 1 over f(x)

A common error is to think the inverse function will be the reciprocal of the function because of the “power of -1 ” notation.

Eg if f(x)=3x, an incorrect expression for f^{-1} would be (3x)^{-1} or \frac{1}{3x} .

### Practice inverse function questions

1. If f(x)=4x-9, find f^{-1}(x):

\frac{x-9}{4} \frac{x+9}{4} \frac{x}{4}+9 \frac{x}{4}-9 Rearranging the formula, first add 9, then divide by 4.

2. If g(x)=\frac{x}{3}+7, find g^{-1}(x):

3x-21 3x-7 3x+21 3x+7 Rearranging the formula, first subtract 7, then multiply by 3.

3. If h(x)=5-x, find h^{-1}(x) :

x-5 x+5 5x 5-x Rearranging the formula, first add y to both sides, then subtract x from both sides.

4. If f(x)=x^3-1, find f^{-1}(x):

\sqrt{x}+1 \sqrt{x+1} x3+1 \sqrt{x-1} Rearranging the formula, first add 1, then cube root.

5. If g(x)=\sqrt{x}+6, find g^{-1}(x) :

x^2+36 x^2-6 x^2-12x+36 x^2+6 Rearranging the formula, first subtract 6, then square and expand the brackets.

6. If h(x)=\frac{2x}{x+3}, find h^{-1}(x) :

\frac{3x}{2+x} \frac{3x}{x-2} \frac{2x}{x-3} \frac{3x}{2-x} See working and use example 4 to help.

\begin{aligned} & \frac{2y}{y+3}=x \\\\ & 2y=x\left( y+3 \right) \\\\ & 2y=xy+3x \\\\ & 2y-xy=3x \\\\ & y\left( 2-x \right)=3x \\\\ & y=\frac{3x}{2-x} \\\\ & {{h}^{-1}}\left( x \right)=\frac{3x}{2-x} \end{aligned}

### Inverse functions GCSE questions

1. (a) If f(x)=6x-5, find f^{-1}(x).

(b) If g(x)=3+\sqrt{x}, find g^{-1}(x).

(4 marks)

(a)

Attempt the rearrange formula 6y-5=x

(1)

\frac{x+5}{6}

(1)

(b)

Attempt the rearrange formula 3+\sqrt{y}=x

(1)

(x-3)^2   or  x^2-6x+9

(1)

2. If f(x)=3x-2 and g(x)=4x+7

(a) Find f^{-1}(4)

(b) Find when f(x)=g^{-1}(x)

(6 marks)

(a)

Set 3x-2=4 or find f^{-1}(x)=\frac{x+2}{3}

(1)

(1)

(b)

g^{-1}(x)=\frac{x-7}{4}

(1)

Set equal 3x-2=\frac{x-7}{4}

(1)

Get x’s on both sides of the equation 12x-x=-7+8

(1)

(1)

3.  If f(x)=\frac{x+5}{x+3}, find f^{-1}(x).

(4 marks)

x=\frac{y+5}{y+3}

x(y+3)=y+5   Multiplying by denominator

(1)

xy-y=5-3x   Expand and isolate

(1)

y(x-1)=5-3x   y factored out

(1)

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

(1)

## Learning checklist

You have now learned how to:

• where appropriate, interpret simple expressions as functions with inputs and outputs
• interpret the reverse process as the ‘inverse function’

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