https://file.aiquickdraw.com/mathbot/file/1727049201988z6u0l5nv.jpg

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!

Given f(x) = {(-2,1), (0,2), (4,0)} and g(x) = sqrt(9 - x^2), find f(g(x)).

Learn how to solve the composition of two functions, f(x) = {(-2, 1), (0, 2), (4, 0)} and g(x) = sqrt(9 - x^2). This problem involves square root functions, function composition, and understanding domain restrictions, suitable for students in Grades 9-12.

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