Question:

Step 1: Recognize the relationship between $g(x)$ and $f^{-1}(x)$. We are given that $g(x) = \frac{1}{f^{-1}(x)}$, which can be rewritten as: $$g(x) = [f^{-1}(x)]^{-1}$$ Step 2: Differentiate $g(x)$ using the chain rule. Applying the chain rule to $g(x) = [f^{-1}(x)]^{-1}$: $$g'(x) = -1 \cdot [f^{-1}(x)]^{-2} \cdot \frac{d}{dx}[f^{-1}(x)]$$ $$g'(x) = -\frac{1}{[f^{-1}(x)]^2} \cdot \frac{d}{dx}[f^{-1}(x)]$$ Step 3: Apply the inverse function derivative formula. The derivative of an inverse function is given by: $$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$$ Step 4: Substitute the inverse function derivative formula into the expression for $g'(x)$. $$g'(x) = -\frac{1}{[f^{-1}(x)]^2} \cdot \frac{1}{f'(f^{-1}(x))}$$ Step 5: Find $f^{-1}(4)$ using the given data table. Looking at the table, we need to find the value of $x$ such that $f(x) = 4$. From the data: $f(3) = 4$ Therefore: $f^{-1}(4) = 3$ Step 6: Find $f'(f^{-1}(4))$ using the data table. Since $f^{-1}(4) = 3$, we need $f'(3)$. From the table: $f'(3) = \frac{4}{3}$ Step 7: Calculate $g'(4)$ by substituting all values. $$g'(4) = -\frac{1}{[f^{-1}(4)]^2} \cdot \frac{1}{f'(f^{-1}(4))}$$ $$g'(4) = -\frac{1}{(3)^2} \cdot \frac{1}{\frac{4}{3}}$$ $$g'(4) = -\frac{1}{9} \cdot \frac{3}{4}$$ $$g'(4) = -\frac{3}{36} = -\frac{1}{12}$$ **Final Answer:** The value of $g'(4) = -\frac{1}{12}$, which corresponds to **Option 1**.