To find $f(g(x))$, also known as the composition of functions $f \circ g$, we substitute the outputs of $g(x)$ into the function $f(x)$.

Given the sets of points for $f$ and $g$:

• $f = \{(-4, 2), (-1, 4), (3, 0)\}$
• $g = \{(-3, 3), (1, 3), (2, 0), (4, -1)\}$

We need to follow these steps:

1. Identify the value of $g(x)$ for each $x$ in the domain of $g$.
2. Substitute these $g(x)$ values into $f(x)$, checking if $f$ is defined for those values.

Step 1: Find $g(x)$

• For $x = -3$, $g(-3) = 3$.
• For $x = 1$, $g(1) = 3$.
• For $x = 2$, $g(2) = 0$.
• For $x = 4$, $g(4) = -1$.

Step 2: Find $f(g(x))$

Now, we substitute the values of $g(x)$ into $f(x)$:

• For $g(-3) = 3$, we find $f(3)$. From the definition of $f$, $f(3) = 0$, so $f(g(-3)) = 0$.
• For $g(1) = 3$, we find $f(3)$ again. Thus, $f(g(1)) = 0$.
• For $g(2) = 0$, we check if $f(0)$ is defined, but $0$ is not in the domain of $f$, so $f(g(2))$ is undefined.
• For $g(4) = -1$, we find $f(-1)$. From the definition of $f$, $f(-1) = 4$, so $f(g(4)) = 4$.

Conclusion:

The composition $f(g(x))$ is:

• $f(g(-3)) = 0$
• $f(g(1)) = 0$
• $f(g(2)) = \text{undefined}$
• $f(g(4)) = 4$

Would you like further details on function composition?

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Relative questions:

1. How do you determine the domain of a composite function?
2. What happens when a value in $g(x)$ is not in the domain of $f(x)$?
3. Can you graph $f \circ g$ to visualize its behavior?
4. How would $f \circ g$ change if a new point were added to either function?
5. What is the significance of an undefined value in function composition?

Tip: Always check if the output of $g(x)$ lies within the domain of $f(x)$ before evaluating the composite function.