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The question you're working on deals with a piecewise function \( f(x) \) defined as follows:

\[
f(x) = 
\begin{cases} 
3x & \text{if } 0 \leq x < 2, \\
x + 3 & \text{if } 2 \leq x \leq 4.
\end{cases}
\]

Let's go through each part of the question.

### (a) **Determine the domain of \( f \):**
The function is defined for \( 0 \leq x \leq 4 \), covering both intervals. Therefore, the domain of \( f \) is:
\[
\boxed{[0, 4]}.
\]

### (b) **Evaluate \( f(-1), f(1), f(5):**

- **\( f(-1) \)**: Since \( x = -1 \) is outside the domain \( [0, 4] \), \( f(-1) \) is **undefined**.
- **\( f(1) \)**: For \( 0 \leq 1 < 2 \), the rule is \( f(x) = 3x \). So, \( f(1) = 3 \times 1 = 3 \).
- **\( f(5) \)**: Since \( x = 5 \) is outside the domain \( [0, 4] \), \( f(5) \) is **undefined**.

### (c) **Graph \( f \):**
The correct graph (labeled **D**) consists of two line segments:
- From \( (0, 0) \) to \( (2, 6) \), using the rule \( f(x) = 3x \).
- From \( (2, 5) \) to \( (4, 7) \), using the rule \( f(x) = x + 3 \).

### (d) **Is \( f \) continuous on its domain?**
\( f \) is **not continuous** at \( x = 2 \), because:
- The left-hand limit at \( x = 2 \) (from the first interval) is \( 6 \) (since \( 3 \times 2 = 6 \)).
- The right-hand limit at \( x = 2 \) (from the second interval) is \( 5 \) (since \( 2 + 3 = 5 \)).

Since these limits do not match, there is a jump discontinuity at \( x = 2 \).

Do you need further details on any part?

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### 5 Related Questions:
1. What is a piecewise function and how is it graphed?
2. How can you determine if a function is continuous at a point?
3. What are the conditions for a function to be continuous on an interval?
4. How do you evaluate limits for piecewise functions?
5. What are the different types of discontinuities in functions?

### Tip:
When working with piecewise functions, always check the boundaries of each interval carefully to ensure correct evaluation and continuity.

Complete the following for f(x) = 3x if 0 <= x < 2 and f(x) = x + 3 if 2 <= x <= 4. (a) Determine the domain of f. (b) Evaluate f(-1), f(1), and f(5). (c) Graph f. (d) Is f continuous on its domain?

Explore how to handle a piecewise function defined as f(x) = 3x for 0 <= x < 2 and f(x) = x + 3 for 2 <= x <= 4. Learn to determine the domain, evaluate values, graph the function, and assess its continuity. Ideal for Grades 9-11.

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