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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 \( [-4, \infty) \).

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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!

Explore how to find the range of the function f(x) = sqrt(x+4) + x. This solution involves analyzing the function’s domain, increasing nature, and behavior at infinity. Suitable for students in Grades 11-12 studying calculus and algebra.

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