Let’s analyze the problem and answers provided.

Given Function:

The function is $f(x) = \sqrt{9 - x^2}$.

1. Table of Values:

The table is mostly completed with $x$-values and their corresponding $f(x)$-values:

It seems that some values in the table are incorrect because they contradict the function $f(x) = \sqrt{9 - x^2}$. Let me confirm correct outputs for each $x$.

Correct Calculation of $f(x)$:

1. $x = -3.5$: $f(x) = \sqrt{9 - (-3.5)^2} = \sqrt{9 - 12.25} \quad (\text{undefined since the square root of a negative number does not exist!})$ Hence, $f(x)$ is undefined at $x = -3.5$.

2. $x = -3$: $f(x) = \sqrt{9 - (-3)^2} = \sqrt{9 - 9} = \sqrt{0} = 0.$

3. $x = -1$: $f(x) = \sqrt{9 - (-1)^2} = \sqrt{9 - 1} = \sqrt{8} \approx 2.83.$

4. $x = 0$: $f(x) = \sqrt{9 - 0^2} = \sqrt{9} = 3.$

5. $x = 1$: $f(x) = \sqrt{9 - 1^2} = \sqrt{9 - 1} = \sqrt{8} \approx 2.83.$

6. $x = 3$: $f(x) = \sqrt{9 - 3^2} = \sqrt{9 - 9} = \sqrt{0} = 0.$

7. $x = 3.5$: As with $x = -3.5$, $f(x)$ is undefined because: $f(x) = \sqrt{9 - 3.5^2} = \sqrt{9 - 12.25}.$

Corrected Table:

2. Values of $x$ without Outputs:

The function $f(x)$ is only defined when the expression inside the square root is non-negative: $9 - x^2 \geq 0 \implies x^2 \leq 9.$ This gives $x \in [-3, 3]$. Any value of $x$ outside this range (e.g., $x = -3.5$ or $x = 3.5$) does not have an associated output because the square root of a negative number is undefined.

3. Values of $x$ that DO have Outputs:

As stated above, $x$-values must satisfy $-3 \leq x \leq 3$.

4. Highest and Lowest $y$-Values:

From the corrected table and function:

• The highest $y$-value is 3 (at $x = 0$).
• The lowest $y$-value is 0 (at $x = \pm 3$).

Do you need further clarification on any part of this solution?

Here are some follow-up questions to deepen understanding:

1. Why is the domain restricted to $x \in [-3, 3]$?
2. How do we calculate the range of $f(x) = \sqrt{9 - x^2}$?
3. What would happen to the graph if we replaced 9 with another constant (e.g., 16)?
4. How does the square root function impact the shape of the graph?
5. Can you find the equation of the graph in standard form?

Tip: When working with square root functions, always check the domain where the expression inside the square root is non-negative.