Inverse Functions and their derivatives in a nutshell

In Math, Inverse Function is a function whose relation to a given function is such that their composite is the identity function. Inverse Function can also be defined as the function obtained by expressing the dependent variable of one function as the independent variable of another; for instance f and g are inverse-functions if f(x)=y and f(y)=x. It is denoted by (-1) in the power, inverse of f(x) is f-1(x).

One important point we need to remember is that the inverse of a function may not always be a function. While finding Inverse of Function of a function y in terms of x, we just switch the x and y and then solve for y. The new function [y-1] we get is the inverse-function of the given original function. For instance, finding Inverse of a Function f(x)= (x/3) -1 involves a list of steps. First let us take f(x)=y=(x/3) -1. In the next step we switch the x and y, that gives us, x = (y/3)-1. The last step would be to solve for y, finally we get y= y-1= 3x+3 which is the inverse-function of f(x) = (x/3) -1

There are four main steps involved in Solving Inverse Functions. Given a function f(x) in terms of x,
1. First step, we write the given function f(x) equal to y
2. Second step involves interchanging the x and y
3. In the third step, we solve for y
4. Fourth step, we write y as f-1(x) which is the inverse-function of f(x)

Considering an example, let us now learn how to solve Inverse functions using the above steps. Given the function f(x) = 2x-3, let us solve inverse-function [f-1(x)]. The first step would be to write y = 2x-3. Next we interchange x and y which gives, x = 2y-3. Now, we solve for y; y = (x+3)/2. The last step is to put f-1(x) in place of y which gives us f-1(x)= (x+3)/2  which is the inverse-function of the given function.

Derivative of an Inverse Function, if for a function f an inverse-function f-1 exists and if f is differentiable at f-1(x) and f’[f-1(x)] is not equal to zero, then f-1 is differentiable at x and the formula is given as:

Considering an example, let us now learn how to solve Inverse functions using the above steps. Given the function f(x) = 2x-3, let us solve inverse-function [f-1(x)]. The first step would be to write y = 2x-3. Next we interchange x and y which gives, x = 2y-3. Now, we solve for y; y = (x+3)/2. The last step is to put f-1(x) in place of y which gives us f-1(x)= (x+3)/2  which is the inverse-function of the given function.

Derivative of an Inverse Function, if for a function f an inverse-function f-1 exists and if f is differentiable at f-1(x) and f’[f-1(x)] is not equal to zero, then f-1 is differentiable at x and the formula is given as:

A square root function is the inverse-function of a square function f(x) = x2. So, the derivative of an Inverse Function of square function f(x) = x2 can be solved using the power rule