Question

Verify that f has an inverse. 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.) function real number f(x) = x + 2 x − 6 , x > 6 a = 3

Answer 1

we have that

f(x) = x + 2 x − 6------------ > f(x) = 3x− 6
The function is continuous in the interval (-∞,∞)

y=3x-6--------- >3x=y+6------> x=(y+6)/3

Interchange x and y to find the inverse function
(f −1)(x)=(x+6)/3----> (x/3)+2
Apply derivative with respect to x
(f −1)'(x)=1/3
for all values of x (f −1)'(x)=1/3

therefore
for a=3 (f −1)'(a)=1/3