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We are given the function \( h(x) = \sqrt{-x + 3} \), and we need to find its range on the domain \( [-1, 0] \).

### Step 1: Analyze the domain
For \( h(x) = \sqrt{-x + 3} \), the expression inside the square root must be non-negative:
\[
-x + 3 \geq 0 \implies x \leq 3.
\]
Thus, \( x \) can take values up to 3. However, the domain is restricted to \( [-1, 0] \), so we will only consider this interval.

### Step 2: Evaluate the function at the domain boundaries
We will now compute the values of \( h(x) \) at the endpoints of the domain:

- At \( x = -1 \):
  \[
  h(-1) = \sqrt{-(-1) + 3} = \sqrt{1 + 3} = \sqrt{4} = 2.
  \]
- At \( x = 0 \):
  \[
  h(0) = \sqrt{-(0) + 3} = \sqrt{3}.
  \]

Thus, at \( x = -1 \), \( h(x) = 2 \), and at \( x = 0 \), \( h(x) = \sqrt{3} \).

### Step 3: Determine the range
Since the function \( h(x) \) is decreasing in the interval \( [-1, 0] \), the range will consist of values between \( h(0) = \sqrt{3} \) and \( h(-1) = 2 \).

Hence, the range of \( h(x) \) on the domain \( [-1, 0] \) is:
\[
[\sqrt{3}, 2].
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does changing the domain affect the range of a square root function?
2. What happens to the graph of the function if the term inside the square root changes?
3. Can the range of a square root function ever be negative?
4. How do we determine the range of a more complex function like \( f(x) = \sqrt{-2x + 5} \)?
5. What is the difference between the domain and range of a function?

**Tip:** For square root functions, always check that the expression inside the root is non-negative to identify valid domains.

Find the range of the function h(x) = √(-x + 3) on the domain [-1, 0].

Learn how to find the range of the square root function h(x) = √(-x + 3) on the domain [-1, 0]. This step-by-step solution covers key algebraic concepts such as analyzing the domain and computing the range of a square root function.

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