Consider the function for f (x)

Question

Consider the function  for f (x) = 9/(x-6) for x > 6 .



(A)

Find  f^-1(7) = ?

B) Use Theorem 7, page 156 of the Stewart Essential Calculus textbook to find  (f^-1) (7) (f^-1)' (7) = ?

Answer 1

If f(a) = b then f^-1(b) = a

We first need to find the value of x for which f(x) = 7.

9 / (x-6) = 7

7x - 42 = 9

7x = 51

x = 51/7 So, f^-1(7) = 51/7

[f^-1(x)]' = 1 / [f'(f^-1(x))] Note that f'(x) = -9 / (x-6)²

So, [f^-1(7)]' = 1 / [f'(51/7)] = -(51/7 - 6)² / 9 = - (9/7)² / 9 = -9 / 49