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Let's go over this function and the required calculations step by step. The function \( h(x) \) is defined piecewise as:

\[
h(x) =
\begin{cases} 
5x + 10 & \text{if } x < 0 \\
-3x + 6 & \text{if } 0 \leq x < 8 
\end{cases}
\]

### A) The value of \( h(-10) \)
Since \( -10 < 0 \), we use the first piece of the function: \( h(x) = 5x + 10 \).

Substituting \( x = -10 \):

\[
h(-10) = 5(-10) + 10 = -50 + 10 = -40
\]

So, the correct answer is that \( h(-10) = -40 \), corresponding to the first choice provided, where \( 5 \cdot (-10) + 10 \) is correctly calculated.

---

### B) \( h(0) \)
For \( x = 0 \), since \( 0 \leq 0 < 8 \), we use the second piece of the function: \( h(x) = -3x + 6 \).

Substituting \( x = 0 \):

\[
h(0) = -3(0) + 6 = 0 + 6 = 6
\]

Thus, \( h(0) = 6 \).

---

### C) \( h(1) \)
For \( x = 1 \), since \( 0 \leq 1 < 8 \), we again use the second piece of the function: \( h(x) = -3x + 6 \).

Substituting \( x = 1 \):

\[
h(1) = -3(1) + 6 = -3 + 6 = 3
\]

So, \( h(1) = 3 \).

---

### D) The value of \( h(11) \)
The function \( h(x) \) is only defined for \( x < 8 \). Since \( x = 11 \) is outside this domain, the function is **not defined** for \( h(11) \).

The correct answer here is the first option: "is not defined because 11 is not in the domain of \( h \)."

---

Would you like further details on any of these calculations or explanations?

### Related Questions:
1. What is \( h(x) \) for values between 0 and 8, and how does the slope of the function affect these values?
2. How can we graph the function \( h(x) \) to better understand its behavior?
3. Why is the function \( h(x) \) not defined for values greater than or equal to 8?
4. Can we extend the function \( h(x) \) to be defined for values outside of \( x < 8 \)?
5. What does it mean for a function to be piecewise, and how is it commonly used in real-world applications?

**Tip:** Always double-check the domain of piecewise functions to ensure you’re using the correct rule for the given \( x \)-value.

Calculate the following values using the given function: h(x) = 5x + 10 if x < 0, h(x) = -3x + 6 if 0 <= x < 8.

Learn how to calculate values for the piecewise function h(x) defined as h(x) = 5x + 10 if x < 0 and h(x) = -3x + 6 if 0 ≤ x < 8. Step-by-step solutions for h(-10), h(0), h(1), and determining if h(11) is defined.

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