### Inverse Functions

When a function is both one-one and on-to then the function can have an inverse. In other words, a function with A as domain and B as co-domain. If each member of A is associated with each member of B and no two members of B is associated with one member of A and no two members of A is associated with one member of B then the function is said to have an inverse.

The function y = 2x + 1 has its domain as well as range as R.
The inverse function is x = (y - 1)/2. This function also has its range and domain as R.

If a given function is not one-one on its domain, we can choose a subset of the domain on which it is one-one, and then define its inverse function.

The trigonometric functions sine is not one-one but part of its domain from -π/2 to π/2 is one-one, hence we can define this subset of the domain and find the inverse of the function with domain -1 to 1. Similarly other functions can be determined.

FunctionInverse Function
y = exx = log y
y = axx = loga y
y = sin xx = sin-1 y
y = cos xx = cos-1 y
y = tan xx = tan-1 y
y = cosec xx = cosec-1 y
y = sec xx = sec-1 y
y = cot xx = cot-1 y
y = 4x + 3x = (y - 3)/4

The graph of a function and its inverse is symmetrical in the line y = x.