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We are tasked with finding \( h'(1) \), where \( h(x) = f(g(x)) \). This is the composition of two functions: \( g(x) = 2x + 1 \) and the piecewise linear function \( f(x) \) as shown in the graph.

### Step 1: Differentiate \( h(x) = f(g(x)) \)
By the chain rule, we know that:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
where \( g'(x) \) is the derivative of \( g(x) \), and \( f'(x) \) is the derivative of \( f(x) \).

### Step 2: Calculate \( g'(x) \)
The function \( g(x) = 2x + 1 \) is linear, so:
\[
g'(x) = 2
\]

### Step 3: Evaluate \( g(1) \)
Substitute \( x = 1 \) into \( g(x) \):
\[
g(1) = 2(1) + 1 = 3
\]
Thus, \( g(1) = 3 \), so we need to evaluate \( f'(3) \).

### Step 4: Determine \( f'(3) \)
Now, let's look at the graph of \( f(x) \). The graph consists of two line segments:
- From \( (0, 0) \) to \( (1, 4) \)
- From \( (1, 4) \) to \( (4, -2) \)

The slope of the segment from \( (1, 4) \) to \( (4, -2) \) is:
\[
\text{slope of } f(x) \text{ on this segment} = \frac{-2 - 4}{4 - 1} = \frac{-6}{3} = -2
\]
Since \( 3 \) lies on this second segment, we conclude that \( f'(3) = -2 \).

### Step 5: Calculate \( h'(1) \)
Now apply the chain rule:
\[
h'(1) = f'(g(1)) \cdot g'(1) = f'(3) \cdot 2 = (-2) \cdot 2 = -4
\]

Thus, the value of \( h'(1) \) is \( \boxed{-4} \).

Would you like more details on any part of the explanation?

### Related questions:
1. What is the geometric meaning of \( f'(x) \) in this context?
2. How does the chain rule work for function compositions?
3. What are other examples where a piecewise function is involved in differentiation?
4. Can we explore other values of \( h'(x) \) for different points besides \( x = 1 \)?
5. How does the slope of a line segment impact the derivative of a function?

### Tip:
Always check the graph of a piecewise linear function carefully to identify the correct slope (derivative) for the relevant interval.

The graph of the function f, consisting of two line segments, is shown in the figure above. Let g be the function given by g(x) = 2x + 1, and let h be the function given by h(x) = f(g(x)). What is the value of h'(1)?

Learn how to find the derivative of a composite function h(x) = f(g(x)) using the chain rule. This problem covers differentiation of piecewise linear functions, analyzing slopes, and applying the chain rule to obtain h'(1). Suitable for Grades 10-12.

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