# Working out the inverse function (inverse functions)

The inverse function f⁻¹(x) is a reflection of f(x) in the mirror line y = x. To work out the inverse function follow these steps:

Step 1: Change f(x) for x and x for f⁻¹(x).

Step 2: Make f⁻¹(x) the subject.

Let’s run through a few examples:

**Example 1**

Work out the inverse function if f(x) = 9x + 1

Step 1: Change f(x) for x and x for f⁻¹(x).

f(x) = 9x + 1

x = 9f⁻¹(x) + 1

Step 2: Make f⁻¹(x) the subject.

x = 9f⁻¹(x) + 1

Take 1 off both sides

x – 1 = 9f⁻¹(x)

Divide both sides by 9.

(x-1)/9 = f⁻¹(x)

So our final inverse function is:

f⁻¹(x) = (x-1)/9

**Example 2**

Work out the inverse function if f(x) = 2x - 2

Step 1: Change f(x) for x and x for f⁻¹(x).

f(x) = 2x – 2

x = 2f⁻¹(x) - 2

Step 2: Make f⁻¹(x) the subject.

x = 2f⁻¹(x) - 2

Add 2 to both sides

x + 2 = 2f⁻¹(x)

Multiply both sides by 3.

3x + 6 = f⁻¹(x)

So our final inverse function is:

f⁻¹(x) = 3x + 6

**Example 3**

Work out the inverse function if f(x) = 1/x + 8

Step 1: Change f(x) for x and x for f⁻¹(x).

f(x) = 1/x + 8

x = 1/ f⁻¹(x) + 8

Step 2: Make f⁻¹(x) the subject.

x = 1/ f⁻¹(x) + 8

Take 8 from both sides.

x -8 = 1/f⁻¹(x)

Take the reciprocal of both sides

1/(x-8) = f⁻¹(x)

So our final inverse function is:

f⁻¹(x) = 1/(x-8)

For some more questions on inverse functions try these links:

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