Inverse Functions

Inverse Functions (Definition)

    Let f(x) be a 1-1 function then g(x) is an inverse function of f(x) if

f(g(x)) = g(f(x)) = x

Example:

For 

        f(x) = 2x - 1

        f ^-1(x) = 1/2 x +1/2

    Since 

        f(f ^-1(x) ) = 2[1/2 x +1/2] - 1 = x

    and 

        f ^-1(f(x)) = 1/2 [2x - 1] + 1/2 = x

The Horizontal Line Test and Roll's Theorem

Note that if f(x) is differentiable and the horizontal line test fails then 

        f(a)  =  f(b) 

and Rolls theorem implies that there is a c such that 

        f '(c) = 0

A partial converse is also true:  

    

Theorem If f is differentiable and f '(x) is always non negative (or always non positive) then f(x) has an inverse.

Example:

    f (x) = x³ + x - 4 

has an inverse since

    f'(x) = 3x² + 1 

which is always positive.

Continuity and Differentiability of the Inverse Function 

Theorem f  continuous implies that f -1 is continuous. f  increasing implies that  f -1 is increasing. f decreasing implies that f -1 is decreasing. f differentiable at c and f '(c)  is not 0 implies that f -1 is differentiable at f (c). If g(x) is the inverse of  the differentiable f(x) then                               1  g'(x) =                                 f '(g(x))  if f '(g(x)) is not 0.

Proof  of (5)

Since 

        f (g(x)) = x 

we differentiate implicitly:

        d                             d
               f (g(x))    =             x        
       dx                           dx

    Using the chain rule 

        y = f(u),  u = g(x)

       dy           dy        du                     
                =                     
       dx            du       dx

        =   f '(u)  g'(x)   =   f '(g(x))  g'(x)

    So that

    f '(g(x))  g'(x)   =   1  

Dividing, we get:

Example:  

    For x > 0,  let 

        f(x) = x² 

and 

         

be its inverse, then

    Note that 

       

Exercises:  

1. Let 
   
           f(x) = x³ + x - 4

     Find 
   
           d/dx[f ^-1(-4)]

   
   

2. Let 
   
          

               Find  
   
           d/dx[f ^-1(0)]

   

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