Inverse Functions

Inverse Functions

GCSE(H),

An inverse function reverses the effects of the original function. It is written as f-1(x).

If f(x) = 3x + 2, determine the inverse function for f(x):

The original function is:

x   x3 +2 → 3x+2

Reverse the operations, starting with x:

`frac((x-2))(3)`  ÷3 -2 x ←

Therefore f-1(x) = `frac(x-2)(3)`. If a function is performed on a value; and the inverse function is applied to the result, then the original value is obtained. This is a useful check to ensure that the inverse function has been correctly determined. Check with a random number: use f(3) to obtain a result of 11. Then use the result in the inverse function, so f-1(11) = `frac((11-2))(3)` = 3.

Examples

1. Find the inverse function of f(x) = 5(x - 1)

Answer: f-1(x) = `frac(x)(5)` + 1

Function: x- 1 x 5 → 5(x -1)

Inverse: `frac(x)(5)` + 1 ←   + 1 ÷ 5 x

Check: use 3 in the original function: 5(3 - 1) = 10. Use 10 in the inverse function: 10 ÷ 5 + 1 = 3.

2. If f(x) = x2 + 5, what is the inverse function?

Answer: √(x - 5)

Function: x x2 + 5 x2 + 5

Inverse: √(x - 5) ← - 5 x

Check: use 3 in the original function: 32 + 5 = 14. Try f-1(14): √(14 - 5) = √9 = 3.