Question

Let f be a differentiable function such that f(3)=15, f(6)=3, f ' (3)= -8, and f ' (6)= -2. The function g is differentiable and g(x)= f inverse (x) for all x. What is the value of g '(3)
A: -1/2B: -1/8C: 1/6D: 1/3E: Value cannot be determined

Answer 1

Answer:

A. $g'(3)=-\frac{1}{2}$

Step-by-step explanation:

We have been given that f be a differentiable function such that$f(3)=15$, $f(6)=3$, $f'(3)= -8$, and $f'(6)= -2$. The function g is differentiable and $g(x)= f^{-1} (x)$ for all x.

We know that when one function is inverse of other function, so:

$g(f(x))=x$

Upon taking derivative of both sides of our equation, we will get:

$g'(f(x))*f'(x) =1$

$g'(f(x))=\frac{1}{f'(x)}$

Plugging $x=6$ into our equation, we will get:

$g'(f(6))=\frac{1}{f'(6)}$

Since $g(x)= f^{-1} (x)$, then $g'(f(6))=g'(3)$.

$g'(3)=\frac{1}{f'(6)}$

Since we have been given that $f'(6)= -2$, so we will get:

$g'(3)=\frac{1}{-2}$

$g'(3)=-\frac{1}{2}$

Therefore, $g'(3)=-\frac{1}{2}$ and option A is the correct choice.