Inverse functions

An inverse function is the opposite of the original function. The Notation of an inverse function is $f^{- 1} (x)$ , where the original function is f (x).

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Last Updated: 01.12.2023
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Only one-to-one Functions (where one value of the domain goes to only one value in the range) can have inverses.

One-to-one Functions are when one value of the domain goes to only one value in the range. This differs from one-to-many functions where one value of the domain can go to several values in the range

Finding the inverse of a function

To find the inverse of a function, you need to

Replace the function Notation with y (eg f (x) become y)
Rearrange the function so that x is the subject
Replace the x with the inverse function notation (eg x becomes $f^{- 1} (x)$ and y with x.

Find the inverse function of f (x) = 5x + 6

Replace the function notation with y: y = 5x + 6
Rearrange the function so that x is the subject: $y - 6 = 5 x$ then $\frac{(y - 6)}{5} = x$
Replace the x with the inverse function notation and y with x: $g^{- 1} (x) = \frac{x - 6}{5}$

Find the inverse function of $j (x) = x^{2} - 6$

Replace the function notation with y: $y = x^{2} - 6$
Rearrange the function so that x is the subject: $y + 6 = x^{2}$ then $\sqrt{y + 6} = x$
Replace the x with the inverse function notation and y with x: $j^{- 1} (x) = \sqrt{x + 6}$

Answering inverse functions questions

There are several types of questions you can be asked involving inverse functions. The questions can ask you to use one or more methods.

Solving the function when x is known

This type of question is shown through $f^{- 1} (x)$ , where x is then replaced with a constant such as, $f^{- 1} (4)$ . To solve these questions, all xs are replaced with the Number in the function.

Solve $f^{- 1} (4) = 6 x - 2$

f^{- 1} (4) = 6 (4) - 2 = 24 - 2 = 22

$f^{- 1} (4) = 22$

Solving the function when it is set to a value

This type of question is shown through $g^{- 1} (x) = y$ . To solve this type of question, you set the function equal to y and then rearrange the question to get x on its own.

When $g^{- 1} (x) = 6 x + 4$ , find x when $g^{- 1} (x) = 58$

$58 = 6 x + 4$

$6 x = 54$

$x = 9$

Working with domains and ranges

You can be asked to find domains and ranges for inverse functions. The domain (set of input values) of the original function will be the range (set of possible output values) of the inverse function. The domain of the inverse function will be the range of the original function.

domain

(Set of input values)

Range

(Set of possible output values)

Original function

Inverse function

Original function

Find the inverse of $h (x) = 3 x^{2} + 4$ with a domain of $x \geq 5$ . State the domain and range of $h^{- 1} (x)$ .

Part 1) Finding the inverse.

$y = 3 x^{2} + 4$
$y - 4 = 3 x^{2}$ , then $\frac{y - 4}{3} = x^{2}$ , then $x = \sqrt{\frac{y - 4}{3}}$
$h^{- 1} (x) = \sqrt{\frac{y - 4}{3}}$

Part 2) Finding the domain and the range of the inverse As stated above, the domain of the original functions is the range of the inverse. Therefore, the range of

h^{- 1} (x)

x \geq 5

. To find the domain of

h^{- 1} (x)

, you can find the range of the original function, so we are going to substitute the value for the domain.

h (5) = 3 (5^{2}) + 4

= 3 (25) + 4

= 79

$x \geq 79$

Range:

x \geq 5

Domain:

x \geq 79

Representing inverse functions graphically

There are two ways of drawing an inverse function:

1) Directly reflect the original function in the line y = x using your transformation of a graph skills.

2) By finding the inverse function and then plotting the x and y coordinates.

Directly reflect the original function in the line y = x

An inverse function is the reflection of the original function in the line y = x, therefore we can use the original line and the line y = x as the line of reflection.

Graphically show the inverse of f (x) = 2x + 4

1) The original function (red) depicted graphically

2) The original function (red) and the line of reflection, y = x (blue)

3) The inverse function (green) is obtained through reflecting the original function (red) in the line of reflection (x = y) (blue).

This method might be a bit more difficult when the original function has a variable raised to a power other than 1; for instance, quadratics $(x^{2})$

Plotting the coordinates of the inverse functions

After finding the inverse function, you can plot the domain and range (meaning the x and y coordinates)

Plot the inverse function $g^{- 1} (x) = {(x - 2)}^{2} - 4$ with the domain $0 \leq x \leq 6$

First, find the y coordinates using the inverse function and the domain and then input the values into a table.
x 0 1 2 3 $4$ 5 $6$
y 0 -3 -4 -3 0 5 $12$
Then plot the coordinates onto the graph

Image caption

Then, draw the inverse function by drawing a line through all the points and extending through.

Inverse functions - Key takeaways

Inverse functions are the opposite of the original function. Their notation differs from normal functions due to the $^{- 1}$ . Only one-to-one functions are able to have inverse functions.
Inverse functions can be formed by: 1) replacing the function notation with y; 2) rearranging the original function to make x the subject; 3) replacing the x with the inverse function notation and y with x.
When the inverse function is formed, the domain is the original function's range and the range is the domain of the original function.

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Frequently Asked Questions about Inverse functions

What is an inverse function?

The inverse function is the opposite of the original function.

How do you find the inverse of a function?

Replace the function notation (e.g f(x)) with a y; 2) Rearrange the function so that x is on its own; 3) Replace the x with the inverse function notation (e.g. f-1(x)) and the y with an x.

What is an example of an inverse function?

j-1(x)=x+2

Any equation can be the inverse function as long as it has the inverse function notation at the start

How do you solve inverse functions?

When x is known, you substitute the value of x into the function and solve. If x is not known but the inverse function is set to a constant, you rearrange the inverse function to get x on its own.

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