## 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. and are invertible function. In fact they are inverses of each other so that and The graphs of and are 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 all We can however, make it invertible by restricting the domain to so 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 1. Make the subject.

2. Interchange and 3. Replace with For example, if then applying step 1 gives Now apply step 2 to get And finally step 3 to get  