Inverse Functions

## Inverse Functions

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

To determine the inverse of a function, first work out the steps of the original function, remembering to follow the BIDMAS rules. The easiest way to do that is to lay out the operations in a small table, working from left to right, starting with x and building up the expression.

Now, working below that row at the right hand end and, starting with x, work from right to left. For each original operation, carry out the inverse. At the left hand end you will have the inverse operation.

Finally, carry out a check. Choose any number - preferably one that is easy to calculate - and evaluate the function using that number. Taking the answer from that calculation, apply that number to the inverse function. You should get back to your starting number.

## Example 1

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

Work out the steps in the function as a table, working from left to right

 start - 1 x 5 result x → x - 1 → 5(x - 1) → 5(x-1)

Work out the inverse function, by taking the inverse of each operation from the table above, and working right to left:

 result + 1 ÷ 5 start frac(x)(5) + 1 ← frac(x)(5) + 1 ← frac(x)(5) ← x

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

Then use that result (10) in the inverse function: 10 ÷ 5 + 1 = 3.

You should get back to the starting number.

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

## Example 2

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

Work out the steps to get to the function

 start x^2 + 5 result x → x^2 → x^2 + 5 → x^2 + 5

Working from right to left, carry out the inverse operations:

 result sqrt( ) - 5 start sqrt(x - 5) ← sqrt(x - 5) ← x - 5 ← x

Check: Use 3 in the original function: f(3) = 3^2 + 5 = 14

Substitute 14 into the inverse function: f^-1(14): sqrt(14 - 5) = sqrt(9) = 3

You should get the starting number.

Answer: f^(-1)(x) = sqrt(x - 5)