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Inverse operations

Rearranging equations TrigonometryThis topic is relevant for:

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.

**Inverse functions** are functions which reverse or “undo” another function.

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

We have seen how to 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 find an inverse function we need to rewrite the function using y as the unknown variable and set the function equal to x. Then we need to rearrange the function to make y the subject and write the function using the inverse function notation.

E.g.

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

Inverse functions are also used when finding an unknown angle in a triangle using 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 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 .

In order to find an inverse function:

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

Get your free inverse functions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free inverse functions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEInverse 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:

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

**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.**

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

** Write out the expression for the original function using a ** y

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

** Rearrange the equation to make ** y

\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}

** Write your inverse function using the ** f^{-1}

Use g^{-1} because the original function was g.

{{g}^{-1}}\left( x \right)=18-2x

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

** Write out the expression for the original function using a ** y

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

** Rearrange the equation to make ** y

\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.

** Write your inverse function using the ** f^{-1}

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

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

** Write out the expression for the original function using a ** y

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

** Rearrange the equation to make ** y

\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}

** Write your inverse function using the ** f^{-1}

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

**“**1 overf(x) **“**

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

E.g.

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

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[3]{x}+1

\sqrt[3]{x+1}

x3+1

\sqrt[3]{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}

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

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

**(4 marks)**

Show answer

(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)**

Show answer

(a)

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

**(1)**

Answer of 2

**(1)**

(b)

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

**(1)**

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

**(1)**

Get x’s on one side of the equation 12x-x=-7+8

**(1)**

Answer of x=\frac{1}{11}

**(1)**

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

**(4 marks)**

Show answer

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)**

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’

- Inequalities
- Factorising

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Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

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