To analyze the end behavior of the function \( f(x) = \sqrt{x} + 2 \), we need to examine how the function behaves as \( x \) approaches its extreme values: \( x \to \infty \) (as \( x \) gets very large) and \( x \to 0^+ \) (since the square root function is undefined for negative values of \( x \)).

### 1. **As \( x \to \infty \):**
- \( \sqrt{x} \to \infty \), since the square root of a large number becomes increasingly large.
- Therefore, \( f(x) = \sqrt{x} + 2 \to \infty \), meaning that the function grows without bound as \( x \) increases.

Thus, the end behavior as \( x \to \infty \) is \( f(x) \to \infty \).

### 2. **As \( x \to 0^+ \):**
- \( \sqrt{x} \to 0 \), since the square root of a small positive number approaches 0.
- Therefore, \( f(x) = \sqrt{x} + 2 \to 2 \).

Thus, the end behavior as \( x \to 0^+ \) is \( f(x) \to 2 \).

### Summary:
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to 0^+ \), \( f(x) \to 2 \).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the domain of the function \( f(x) = \sqrt{x} + 2 \)?
2. How do you find the derivative of \( f(x) = \sqrt{x} + 2 \)?
3. What is the range of \( f(x) = \sqrt{x} + 2 \)?
4. How does the graph of \( f(x) = \sqrt{x} + 2 \) compare to the graph of \( f(x) = \sqrt{x} \)?
5. What happens to the function \( f(x) = \sqrt{x} + 2 \) if we add a constant, say \( f(x) = \sqrt{x} + c \)?

**Tip**: To understand end behavior, always consider the limits of the function as \( x \) approaches infinity or zero.

Discover the end behavior of the function f(x) = √x + 2. Understand how the function behaves as x approaches infinity and as x approaches zero. This solution covers key mathematical concepts such as square root functions and end behavior, suitable for Grades 9-11.

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