In this problem, we’re asked to find $g'(0)$ using the information provided. Here’s a breakdown of the solution and why we end up multiplying by $\frac{1}{2}$.

Given Information and Setup

1. We know that $f(g(x)) = \arctan(1 - x)$.
2. To find $g'(0)$, we use the chain rule, differentiating both sides of $f(g(x)) = \arctan(1 - x)$ with respect to $x$.

Differentiating Using the Chain Rule

Differentiating $f(g(x)) = \arctan(1 - x)$ with respect to $x$ gives: $f'(g(x)) \cdot g'(x) = \frac{-1}{1 + (1 - x)^2}$

Evaluating at $x = 0$

When $x = 0$, we have:

• $g(0) = 3.1$ (given),
• Therefore, $f(g(0)) = \arctan(1) = \frac{\pi}{4}$, confirming the setup.

Now, plug $x = 0$ into the differentiated equation: $f'(g(0)) \cdot g'(0) = \frac{-1}{1 + (1 - 0)^2} = \frac{-1}{1 + 1} = -\frac{1}{2}$

Solving for $g'(0)$

From the graph of $f(x)$, we can observe or calculate that $f'(3.1) = -\frac{1}{2}$. Substitute this into the equation: $-\frac{1}{2} \cdot g'(0) = -\frac{1}{2}$

To isolate $g'(0)$, divide both sides by $-\frac{1}{2}$: $g'(0) = \frac{-\frac{1}{2}}{-\frac{1}{2}} = 1$

However, upon checking the options provided, there seems to be an inconsistency with this derivation and the provided answer choices. This indicates a possible interpretation or graph-checking error.