## Inverse Functions

If a function is one to one, so that every value in the domain is associated with a unique value in the codomain and every value in the codomain is associated with a unique value in the domain, then the function can be inverted.

An invertible function can have no turning points (minima or maxima). The gradient must always have the same sign, and if it is zero, can only be zero at a single point on any interval. That is, the function must be always either increasing or decreasing.

andare invertible function. In fact they are inverses of each other so thatandThe graphs ofandare shown below. Each is the reflection of the other in the line

We can often make a non – invertible function invertible by restricting the domain. For example,is non invertible because it is not one to one:for allWe can however, make it invertible by restricting the domain toso that the function is one to one on that domain, then it will be invertible with that restriction.

There is a procedure for finding the inverse of a one to one function

Makethe subject.

Interchangeand

Replacewith

For example, ifthen applying step 1 gives

Now apply step 2 to get

And finally step 3 to get