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.