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The function \( C(x) \) is defined as a piecewise function:

\[
C(x) = \begin{cases} 
300, & \text{if } 0 < x \leq 8 \\
20x + 140, & \text{if } 8 < x \leq 15 \\
15x + 215, & \text{if } x > 15 
\end{cases}
\]

### Domain:
The domain of \( C(x) \) is \( (0, \infty) \), meaning \( x \) can take any value greater than 0.

### Range:
The range is determined by the output values of each piece of the function:

- For \( 0 < x \leq 8 \), \( C(x) = 300 \).
- For \( 8 < x \leq 15 \), \( C(x) \) increases linearly from \( 300 + 20(8 - 8) = 300 \) to \( C(15) = 20(15) + 140 = 440 \).
- For \( x > 15 \), \( C(x) = 15x + 215 \), which produces values greater than 440.

Thus, the range is \( [300, \infty) \).

Let me know if you'd like more details or clarifications!

Here are 5 related questions to explore further:
1. What is the interpretation of each segment of the piecewise function?
2. How can we graph this piecewise function for better visualization?
3. What happens to the slope of the function at each breakpoint?
4. How does the function behave for very large values of \( x \)?
5. Can we express the range more precisely for a bounded domain?

**Tip:** In piecewise functions, it's crucial to check continuity and how the slopes change between segments.

What is the specific domain and range of the piecewise function shown?

Discover how to find the domain and range of a piecewise function that consists of linear segments. This solution breaks down each segment's behavior and explains how to interpret the domain (0, ∞) and range [300, ∞). Suitable for students in Grades 9-11.

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