if g(x)=f^(-1)(x), with f(3)=4 and g(7)=4, then g(4) equals -1/f(7) 1/f(3) -1/f(3) 1/f(7)

Answer:

B. $\frac{1}{f'(3)}$

Explanation:

Step1: Recall inverse - function derivative formula

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

Step2: Substitute $x = 4$ into the formula

We want to find $g'(4)$. By the formula $g'(4)=\frac{1}{f'(g(4))}$.

Step3: Use the given information

We know that $f(3)=4$, so $g(4) = 3$ (since $g$ is the inverse of $f$).

Step4: Calculate $g'(4)$

Substituting $g(4)=3$ into $g'(4)=\frac{1}{f'(g(4))}$, we get $g'(4)=\frac{1}{f'(3)}$.