https://file.aiquickdraw.com/mathbot/file/1728006909566bhdxcjrj.jpg

The function given in the image is:

\[
f(x) = \sqrt{x + 5} - 3
\]

To find the range of this function using graphing technology, you can analyze the behavior of the function step-by-step.

### Steps to find the range:
1. **Domain**: The square root function \(\sqrt{x + 5}\) requires that the expression inside the square root must be non-negative. This means:
   \[
   x + 5 \geq 0 \quad \Rightarrow \quad x \geq -5
   \]
   Therefore, the domain of the function is \(x \geq -5\).

2. **Minimum Value of the Function**: 
   - At \(x = -5\), the function simplifies to:
     \[
     f(-5) = \sqrt{-5 + 5} - 3 = \sqrt{0} - 3 = -3
     \]
     So, the minimum value of \(f(x)\) is \(-3\).

3. **Behavior as \(x \to \infty\)**: 
   - As \(x\) increases, \(\sqrt{x + 5}\) increases without bound, so the function \(f(x) = \sqrt{x + 5} - 3\) also increases without bound.

4. **Conclusion**: 
   - The range of the function is all values starting from \(-3\) to infinity. In interval notation, the range is:
     \[
     \text{Range of } f(x): [-3, \infty)
     \]

Would you like further details on graphing or clarification on this process?

### Related questions:
1. How do you find the domain of functions involving square roots?
2. What is the impact of a horizontal shift on the range of a function?
3. How does adding or subtracting constants outside a square root affect the graph of a function?
4. How can you confirm the range of a function using graphing tools like Desmos?
5. Can the range of this function be influenced by any other transformations?

**Tip**: When dealing with square root functions, always first check the domain, as it restricts the range directly.

Use graphing technology to find the range of the function f(x) = √(x + 5) - 3.

Learn how to find the range of the function f(x) = √(x + 5) - 3 using graphing technology. This problem involves analyzing the domain and range of square root functions and is suitable for students in Grades 9-12.

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