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)`