if g(x)=f^(-1)(x), with f(4)=8 and g(2)=8, then g(8) equals 1/g(4) 1/g(2) 1/f(2) 1/f(4)

Answer:

D. $\frac{1}{f'(4)}$

Explanation:

Step1: Recall inverse - function derivative formula

If $g(x)=f^{-1}(x)$, then $g'(x)=\frac{1}{f'(g(x))}$.

Step2: Substitute $x = 8$ into the formula

We want to find $g'(8)$. By the formula $g'(x)=\frac{1}{f'(g(x))}$, when $x = 8$, we have $g'(8)=\frac{1}{f'(g(8))}$.

Step3: Use the property of inverse functions

Since $g(x)=f^{-1}(x)$, if $f(4)=8$, then $g(8)=4$.

Step4: Get the final result

Substituting $g(8)=4$ into $g'(8)=\frac{1}{f'(g(8))}$, we get $g'(8)=\frac{1}{f'(4)}$.