The image appears to be a part of a calculus or algebra problem evaluating a function $f(x) = \sqrt{x+4} + x$, followed by evaluating the function at specific values for $x$ and determining the range.

To determine the range of the function $f(x) = \sqrt{x+4} + x$, let's analyze it step-by-step.

Step 1: Domain of the function

The function involves a square root $\sqrt{x+4}$, which is defined only when $x+4 \geq 0$ (i.e., $x \geq -4$).

Thus, the domain of the function is $x \in [-4, \infty)$.

Step 2: Behavior of the function

The function is $f(x) = \sqrt{x+4} + x$. To analyze the range, we consider:

• As $x = -4$, $f(-4) = \sqrt{-4+4} + (-4) = 0 - 4 = -4$.
• As $x \to \infty$, the term $\sqrt{x+4}$ grows without bound, and so does $x$, meaning $f(x) \to \infty$.

Step 3: Increasing nature

Since both $\sqrt{x+4}$ and $x$ increase as $x$ increases, $f(x)$ is an increasing function on the interval $x \in [-4, \infty)$.

Step 4: Conclusion

The function starts at $f(-4) = -4$ and increases without bound. Therefore, the range of the function is $[-4, \infty)$.

Would you like further clarification or details? Here are five related questions to explore:

1. How would the domain change if the function had a different form, such as $f(x) = \sqrt{x-2} + x$?
2. What is the derivative of $f(x) = \sqrt{x+4} + x$ and what does it tell us about the function’s slope?
3. How can you verify the range of a function graphically?
4. How does the behavior of square root functions impact the overall function growth?
5. What are other common types of functions where domain and range need careful consideration?

Tip: When determining the range, always check the behavior of the function at the boundaries of the domain and any critical points!