To determine the correct answer, let's analyze each option by evaluating the function $f(x) = -\sqrt{x + 4} - 3$ at specific points.

Step 1: Rewrite the Function

The function is: $f(x) = -\sqrt{x + 4} - 3$

Step 2: Evaluate the Function at Specific Values

1. $f(-3)$: $f(-3) = -\sqrt{-3 + 4} - 3 = -\sqrt{1} - 3 = -1 - 3 = -4$

2. $f(-4)$: $f(-4) = -\sqrt{-4 + 4} - 3 = -\sqrt{0} - 3 = 0 - 3 = -3$

3. $f(0)$: $f(0) = -\sqrt{0 + 4} - 3 = -\sqrt{4} - 3 = -2 - 3 = -5$

4. $f(5)$: $f(5) = -\sqrt{5 + 4} - 3 = -\sqrt{9} - 3 = -3 - 3 = -6$

Step 3: Check Each Statement

• Option A: $f(-3) = f(-4)$ $f(-3) = -4 \quad \text{and} \quad f(-4) = -3$ This is false since $-4 \neq -3$.

• Option B: $f(-3) > f(-4)$ $f(-3) = -4 \quad \text{and} \quad f(-4) = -3$ This is false since $-4 < -3$.

• Option C: $f(0) > f(-3)$ $f(0) = -5 \quad \text{and} \quad f(-3) = -4$ This is false since $-5 < -4$.

• Option D: $f(0) > f(5)$ $f(0) = -5 \quad \text{and} \quad f(5) = -6$ This is true since $-5 > -6$.

Conclusion

The correct answer is: D. $f(0) > f(5)$

Would you like further explanation or details on any of these calculations?

Follow-up Questions:

1. How do we determine if a function is increasing or decreasing over an interval?
2. What is the effect of negative square roots on function values?
3. How would the answer change if the function had a positive square root term?
4. Why does the value of $f(x)$ decrease as $x$ increases in this function?
5. Can we use derivatives to analyze the behavior of this function?

Tip:

When working with functions involving square roots, check the domain to ensure values under the square root are non-negative.