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We are given the function \( h(x) = -3x + 4 \) with the domain \( -2 \leq x < 5 \), and we need to find its range.

### Step 1: Find the values of \( h(x) \) at the endpoints of the domain.

1. **At \( x = -2 \):**
   \[
   h(-2) = -3(-2) + 4 = 6 + 4 = 10
   \]
   So, \( h(x) = 10 \) when \( x = -2 \).

2. **At \( x = 5 \) (not included, but needed to find the boundary):**
   \[
   h(5) = -3(5) + 4 = -15 + 4 = -11
   \]
   So, \( h(x) = -11 \) as \( x \) approaches 5 from the left.

### Step 2: Determine the range.
The function \( h(x) = -3x + 4 \) is a linear function with a negative slope, meaning it decreases as \( x \) increases. Therefore, as \( x \) ranges from \(-2\) to just below 5, \( h(x) \) ranges from \( 10 \) to just above \( -11 \).

Hence, the range of \( h(x) \) is:
\[
-11 < h(x) \leq 10
\]

Thus, the correct answer is the second option: \( -11 < h(x) \leq 10 \).

Would you like more details on any part of the solution?

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Here are 5 related questions to expand on this concept:

1. How do you find the range of a quadratic function on a given domain?
2. What is the general method for finding the range of a function over an interval?
3. How does the slope of a linear function affect its range?
4. Can a linear function have a range that includes infinity?
5. How would the range change if the domain of \( h(x) \) were extended to \( -3 \leq x \leq 6 \)?

**Tip:** Always check the behavior of the function at the endpoints of the domain to determine its range correctly.

What is the range (or image) of the function h(x) = -3x + 4 when -2 ≤ x < 5?

Learn how to find the range of the linear function h(x) = -3x + 4 with the domain -2 ≤ x < 5. This problem covers key algebraic concepts, including how the slope affects the range of a function. Suitable for students in Grades 9-11.

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