To work out the inverse function all you need to do is make x the subject of the function. Once this is done interchange the x with f⁻¹(x) and y with x. (note: f⁻¹(x) is the inverse function)

A few other observations you can make about the inverse function is that it’s a reflection of f(x) in mirror line y = x. Therefore the domain of f(x) becomes the range in f⁻¹(x) and the range of f(x) becomes the domain in f⁻¹(x).

Let’s take a look at a few example on working out f⁻¹(x).

**Example 1**

If f(x) = 5x -7 work out the inverse function.

Firstly you need to make x the subject:

5x -7 = f(x)

Add 7 to both sides

5x = f(x) + 7

Divide both sides by 5

x = (f(x) +7)/5

Now change x to f⁻¹(x) and change f(x) to x.

So our final answer is:

f⁻¹(x) = (x+7)/5

**Example 2**

Work out f⁻¹(x) if f(x) = x/4 -10.

Firstly you need to make x the subject:

x/4 -10 = f(x)

Add 10 to both sides

x/4 = f(x) + 10

Multiply both sides by 4

x = 4f(x) + 40

Now change x to f⁻¹(x) and change f(x) to x.

So our final answer is:

f⁻¹(x) = 4x + 40

**Example 3**

Work out the inverse function if f(x) = for x>0.

Firstly you need to make x the subject:

x² – 6 = f(x)

Add 6 to both sides

x² = f(x) + 6

Take the square root

x = √ (f(x) + 6)

Now change x to f⁻¹(x) and change f(x) to x.

So our final answer is:

f⁻¹(x) = √(x+6)

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