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Function and the  inverse-function:

A function f(x) is said to have  its inverse  say g(x) [ where g(x) = f-1 (x)] , if and only if the f(x) is a one- to one function.

If f(x) and g(x) are inverse functions to each other, then the following conditions are true.

  1. g(f(x))= x
  2.  f(g(x))=x

 

Horizontal line test:

 

If a function passes the horizontal line test, then it is a one-to-one function. Else it is not a one-to-one function and we can not have the inverse to the function.

 

The relation between the derivative of f(x)and the derivative of f-1(x)  :

 

We have the derivative of a function f(x) = f ’(x)

and  the derivative of  inverse function f-1(x) is represented by  [ f-1(x) ]’

Let us derive the relation between these two derivatives.

let  us assume that f-1(x)= g(x).

 

Then as per the inverse properties, we can say that g(f(x)) =x

 

Let us derivate both sides

 \begin{aligned}\dfrac{d}{dx}\left( g\left( f\left( x\right) \right) \right) =\dfrac{dx}{dx}\\  \Rightarrow g'\left( f\left( x\right) \right) \cdot f'\left( x\right) =1\\  \Rightarrow g'\left( f\left( x\right) \right) =\dfrac{1}{f'\left( x\right) }\end{aligned}

 

Therefore, the derivative of the inverse of a function is 1 over the derivative of the function.

 

The derivative of an inverse function Worked out examples.

 

Example1:

 

What is the value of the d/dx [ f-1(x)] when x=2,given that f(x)= x3+x

And f-1(2)=1.

 

 Solution:

Given that  f(x) = x3+x

Therefore the f’ (x) = 3x2+1

Given that the f-1(2)=1 .Therefore f(1) =2.

We have the  formula  g'\left( f\left( x\right) \right) =\dfrac{1}{f'\left( x\right) }

 

To find:

d/dx [ f-1(x)] when x=2

we initially assumed that the f-1(x) = g(x)

hence  d/dx [ f-1(x)] =g’(x).

Therefore we need to find the g’(x) at x=2.

That is the value of g’(2).

From the formula  g'\left( f\left( x\right) \right) =\dfrac{1}{f'\left( x\right) }

On  comparision we can conclude that f(x) =2

And we can observe that f(x) =x3 +x can be 2, if and only if   x=1.

Therefore,

F’(x)= 3x2+1 .Therefore  f’(1) = 3(1)2+1 =4

 

Conclusion :

g’(2) = 1/f’(1)  = 1/4

Hence, the d/dx [ f-1(x)] when x=2 is  1/4.

The derivative of an inverse function at x=2  is 1/4.