If f(x) = x^3 - x - 6 for all real numbers x, and if g is the inverse function of f, then f'(g(0))g'(0) = ?

a) 0
b) 1
c) -1
d) -6
e) (6)^(1/3)

I had a lot of trouble with finding the inverse, If anyone can help me that would be awesome!

Question

If f(x) = x^3 - x - 6 for all real numbers x, and if g is the inverse function of f, then f'(g(0))g'(0) = ?

a) 0
b) 1 
c) -1
d) -6
e) (6)^(1/3)

I had a lot of trouble with finding the inverse, If anyone can help me that would be awesome!

anonymous · Answer

I don't think you need to find the inverse. If f(x) and g(x) are inverses, then
$$g(x)=\frac{ 1 }{ f'(g(x)) }$$
Multiply both sides by $f'(g(x))$ and sub in 0 for x

blacksteel · Answer

You actually don't need to find the inverse function to solve this. It uses the fact that
$$[f^{-1}]'(x) = \frac{ 1 }{ f'(f^{-1}(x))}$$In this case, this means that
$$g'(x) = \frac{ 1 }{ f'(g(x)) }$$So then
$$f'(g(0))g'(0) = \frac{ f'(g(0)) }{ f'(g(0)) } = 1$$

anonymous · Answer

sorry , should read g'(x) =...