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The problem describes a piecewise function \( h(x) \), defined on the interval \([0, 4]\), as follows:

\[
h(x) = 
\begin{cases} 
\sqrt{x + 1}, & \text{if } x \geq 1 \\
x^2 + 2, & \text{if } x \leq 1 
\end{cases}
\]

### Part a: Calculate the values of \( h(x) \) for specific points
- \( h(-1) \)
- \( h(-3) \)
- \( h(0) \)
- \( h(3) \)
- \( h(4) \)
- \( h(8) \)

### Part b: Define the function \( -h \) and sketch the graphs of \( h(x) \) and \( -h(x) \).

Let’s begin by evaluating the function at the requested points:

### Part a) Function evaluations

1. **For \( h(-1) \)**:
   Since \( -1 \leq 1 \), we use the second branch of the function: 
   \[
   h(-1) = (-1)^2 + 2 = 1 + 2 = 3
   \]

2. **For \( h(-3) \)**:
   Since \( -3 \leq 1 \), we use the second branch:
   \[
   h(-3) = (-3)^2 + 2 = 9 + 2 = 11
   \]

3. **For \( h(0) \)**:
   Since \( 0 \leq 1 \), we use the second branch:
   \[
   h(0) = (0)^2 + 2 = 0 + 2 = 2
   \]

4. **For \( h(3) \)**:
   Since \( 3 \geq 1 \), we use the first branch:
   \[
   h(3) = \sqrt{3 + 1} = \sqrt{4} = 2
   \]

5. **For \( h(4) \)**:
   Since \( 4 \geq 1 \), we use the first branch:
   \[
   h(4) = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
   \]

6. **For \( h(8) \)**:
   Since \( 8 \geq 1 \), we use the first branch:
   \[
   h(8) = \sqrt{8 + 1} = \sqrt{9} = 3
   \]

### Part b) Define the function \( -h(x) \)
The function \( -h(x) \) is simply the negative of \( h(x) \), which flips the graph vertically. Therefore:
\[
-h(x) =
\begin{cases} 
-\sqrt{x + 1}, & \text{if } x \geq 1 \\
-(x^2 + 2), & \text{if } x \leq 1 
\end{cases}
\]

This function is obtained by multiplying the values of \( h(x) \) by \( -1 \), making all positive outputs negative.

Would you like a sketch of the graphs for both functions?

---

### Follow-up questions:
1. What happens to the shape of the graph when you apply the negative to the function?
2. Can you identify where \( h(x) \) changes behavior based on the condition \( x \geq 1 \)?
3. How would the graph of \( h(x) \) compare to \( h^{-1}(x) \), the inverse function?
4. What is the range of \( h(x) \) and \( -h(x) \)?
5. Could this piecewise function be continuous? Why or why not?

**Tip:** When graphing piecewise functions, always check the continuity and how the different pieces connect at their boundaries (e.g., \( x = 1 \) in this case).

Given the piecewise function h(x), defined as h(x) = sqrt(x + 1) if x >= 1 and h(x) = x^2 + 2 if x <= 1, calculate specific values for h(x) and define the function -h(x).

Learn how to solve and graph the piecewise function h(x), which combines square root and quadratic terms. Includes step-by-step calculations for specific points and defining -h(x). Suitable for students in Grades 9-12.

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