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: