https://file.aiquickdraw.com/mathbot/file/1728660216726fqlcdp6h.jpg

The image shows a piecewise function \( f(x) \) defined as:

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

Several questions are asked about this function:

### (a) **Determine the domain of \( f(x) \):**
The domain of a function is the set of all \( x \)-values where the function is defined. From the piecewise function, the first part is defined for \( 0 \leq x < 2 \), and the second part is defined for \( 2 \leq x \leq 4 \). Combining these intervals, the domain of \( f(x) \) is:

\[
[0, 4]
\]

So the domain of \( f(x) \) is \( 0 \leq x \leq 4 \).

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

- \( f(-1) \): Since \( -1 \) is not within the domain \( [0, 4] \), \( f(-1) \) is **undefined**.
- \( f(1) \): For \( x = 1 \), use the first part of the function \( 3x \), since \( 1 \) falls in the interval \( 0 \leq x < 2 \). So, \( f(1) = 3(1) = 3 \).
- \( f(5) \): Since \( 5 \) is outside the domain \( [0, 4] \), \( f(5) \) is **undefined**.

### (c) **Graph \( f(x) \):**
To graph the function:

- For \( 0 \leq x < 2 \), plot the line \( y = 3x \).
- For \( 2 \leq x \leq 4 \), plot the line \( y = x + 3 \).

At \( x = 2 \), check the transition: \( f(2) = 2 + 3 = 5 \), so the graph jumps from the value \( 6 \) from the line \( 3x \) to \( 5 \) on the line \( x + 3 \).

### (d) **Is \( f(x) \) continuous on its domain?**
A function is continuous if there are no breaks, jumps, or gaps in the graph. Here, at \( x = 2 \):

- From the first part: \( \lim_{x \to 2^-} f(x) = 3(2) = 6 \).
- From the second part: \( f(2) = 2 + 3 = 5 \).

Since the left-hand limit \( 6 \) does not equal the value at \( x = 2 \) (which is \( 5 \)), the function is **not continuous** at \( x = 2 \). Therefore, \( f(x) \) is **not continuous** on its entire domain.

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Would you like more details on any of these steps? Here are some related questions you might explore:

1. How can I check continuity for more complex piecewise functions?
2. How would I determine the range of \( f(x) \)?
3. Can you explain how to find limits at the transition points of piecewise functions?
4. What is the importance of left-hand and right-hand limits in continuity analysis?
5. How would this function behave if it were defined for all real numbers?

**Tip:** When checking continuity for piecewise functions, always evaluate the limits at the boundary points of the intervals where the function changes definitions.

Complete the following for f(x) = 3x if 0 <= x < 2 and 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?

Learn how to analyze a piecewise function defined as 3x for 0 <= x < 2 and x + 3 for 2 <= x <= 4. This problem covers determining the domain, evaluating the function at specific points, graphing, and testing for continuity on its domain. Suitable for high school students.

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