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.

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`

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)`

See also Inverse Operations

Check out our iOS app: tons of questions to help you practice for your GCSE maths. Download free on the App Store (in-app purchases required).