Verify that f has an inverse function.

Question

Verify that f has an inverse function. Then use the function f and the given real number a to find (f⁻¹)'(a).

 (Hint: See Example 1. If an answer does not exist, enter DNE.)

f(x) =

x − 7

,    x > 7    a = 4

Answer 1

f(x) = √(x-7) f'(x)= 1 / [2√(x-7)]

f'(x) > 0 for x > 7 So, f is increasing when x > 7.

Therefore, f is 1-1 so has an inverse function.

f(23) = √(23-7) = √16 = 4

So, the point (23,4) is on the graph of y = f(x) and the point (4,23) is on the graph of y = f^-1(x).

[f^-1(4)]' = 1 / f'(23) = 1 / (1/8) = 8