inverse functions

Question

Use the function f and the given real number a to find (f -1)'(a)

 

f(x) = (x+7)/(x+3)  , x > −3, a = 2

(f −1)'(a)=??

 

thanks

Answer 1

Suppose that f^-1(2) = b, then f(b) = (b+7)/(b+3) = 2

 

                                                           b+7 = 2b+6

 

                                                               b = 1

 

So, f(1) = 2

 

Now. [f^-1(2)]' = 1/[f'(1)]       f'(x) = [(x+3)-(x+7)]/(x+3)²

 

                                               = -4/(x+3)²

 

                                        So, f'(1) = -4/16 = -1/4

 

Therefore, [f^-1(2)]' = 1/[f'(1)] = -4

 

Answer 2

Switch places with x and y.

 

x = (y + 7) / (y + 3)

 

x(y + 3) = y + 7

 

xy + 3x = y + 7

 

xy - y = 7 - 3x

 

y(x - 1) = 7 - 3x

 

y = (7 - 3x) / (x - 1)

 

f(x) = (7 - 3x) / (x - 1)   ----> inverse

 

 

Take the derivative of this inverse using quotient rule.  Then evaluate derivative when x=2.