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The given function is:

\[
f(x) = \sqrt{0.5x + 2} - 1
\]

### Domain:
For the square root function, the expression inside the square root must be non-negative, meaning:

\[
0.5x + 2 \geq 0
\]

Solve for \(x\):

\[
0.5x \geq -2 \quad \Rightarrow \quad x \geq -4
\]

Thus, the domain is:

\[
x \in [-4, \infty)
\]

### Range:
The square root function, \(\sqrt{0.5x + 2}\), is always non-negative, meaning \(\sqrt{0.5x + 2} \geq 0\). Subtracting 1 from this gives:

\[
\sqrt{0.5x + 2} - 1 \geq -1
\]

Therefore, the minimum value of the function is \(-1\). As \(x\) increases, the function grows without bound. Thus, the range is:

\[
f(x) \in [-1, \infty)
\]

### Correct Answer:
- **Domain:** \((-4, \infty)\)
- **Range:** \([-1, \infty)\)

This corresponds to the last option.

---

Would you like further details or have any questions?

Here are some related questions:
1. How do you find the domain of a square root function in general?
2. What are the properties of the square root function and its transformations?
3. How does the addition or subtraction outside a function affect its range?
4. Why does \( f(x) = \sqrt{0.5x + 2} \) require non-negative values inside the square root?
5. What happens to the range of \( f(x) \) if the function was \( f(x) = -\sqrt{0.5x + 2} - 1 \)?

**Tip:** Always check for domain restrictions when dealing with square root or logarithmic functions!

Determine the domain and range for the following function: f(x) = √(0.5x + 2) - 1

This problem focuses on finding the domain and range of the function f(x) = √(0.5x + 2) - 1. Learn how to calculate the domain by ensuring the expression under the square root is non-negative and determining the function's range based on its behavior.

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