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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.

Explore a calculus problem involving the chain rule to differentiate a composite function f(g(x)) = arctan(1 - x) and find g'(0). This solution is ideal for students in Grades 11-12, covering advanced differentiation techniques and applying the chain rule.

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