The problem in the image asks to find $f(g(x))$, given:

1. $f(x) = \{(-2, 1), (0, 2), (4, 0)\}$
2. $g(x) = \sqrt{9 - x^2}$

Steps:

1. Evaluate $g(x)$:
   $g(x) = \sqrt{9 - x^2}$ is defined for $x \in [-3, 3]$ (since the square root is only defined for non-negative numbers).

2. Substitute $g(x)$ into $f(x)$:
   The values of $f(g(x))$ will depend on the output of $g(x)$, which needs to match one of the input values for $f(x)$, specifically $-2$, $0$, or $4$, since those are the only input values where $f(x)$ is defined.

3. Solve for specific $g(x)$ values:

   • $g(3) = \sqrt{9 - 3^2} = \sqrt{0} = 0$
   • $g(0) = \sqrt{9 - 0^2} = \sqrt{9} = 3$
     This does not match any $x$-values in $f(x)$.
   • $g(2) = \sqrt{9 - 2^2} = \sqrt{5}$ (no match in $f(x)$).

   The only value that matches is when $g(x) = 0$, which corresponds to $f(0) = 2$.

Conclusion:

For values of $x$ where $g(x) = 0$, we get $f(g(x)) = f(0) = 2$. Thus, $f(g(x)) = 2$ when $g(x) = 0$.

Would you like further details on this, or any other specific questions about the process?

Here are some related questions:

1. What is the domain of $g(x) = \sqrt{9 - x^2}$?
2. How do you compose two functions in general, and what does $f(g(x))$ represent?
3. Why does the function $f(x)$ only give values for $x = -2, 0, 4$?
4. What are the other possible inputs for $g(x)$ that could simplify $f(g(x))$?
5. Can we graph both $f(x)$ and $g(x)$ to better visualize their behavior?

Tip: Always check if the values of the inner function are within the domain of the outer function when solving composition problems!