To help you find the domain and range of the composition $f \circ g$ based on the provided mappings, let's analyze the functions from the image.

Step 1: Mapping of $g$

In the diagram, $g$ is the first function. Let's list its mappings:

• $g(1) = 2$
• $g(2) = 4$
• $g(3) = 6$
• $g(4) = 8$

The domain of $g$ is $\{1, 2, 3, 4\}$, and the range of $g$ is $\{2, 4, 6, 8\}$.

Step 2: Mapping of $f$

In the second diagram, $f$ is the second function. Here are its mappings:

• $f(2) = 3$
• $f(4) = 7$
• $f(6) = 0$
• $f(8) = 7$

The domain of $f$ is $\{2, 4, 6, 8\}$, and the range of $f$ is $\{0, 3, 7\}$.

Step 3: Composition $f \circ g$

For the composition $f \circ g$, we apply $g$ first, then $f$ to the result.

• $(f \circ g)(1) = f(g(1)) = f(2) = 3$
• $(f \circ g)(2) = f(g(2)) = f(4) = 7$
• $(f \circ g)(3) = f(g(3)) = f(6) = 0$
• $(f \circ g)(4) = f(g(4)) = f(8) = 7$

Thus, the domain of $f \circ g$ is $\{1, 2, 3, 4\}$, and the range of $f \circ g$ is $\{0, 3, 7\}$.

Answer:

• (a) The domain of $f \circ g$ is $\{1, 2, 3, 4\}$.
• (b) The range of $f \circ g$ is $\{0, 3, 7\}$.

It seems you had the domain partially correct, but you missed $3$, which is also in the domain.

Would you like to go over any other details or need clarification on this? Here are some related questions:

1. What are the key rules for composing two functions?
2. How do you determine the domain of a composition $f \circ g$?
3. Can a function have the same input map to different outputs?
4. What happens if the range of $g$ doesn’t overlap with the domain of $f$?
5. How do you represent a composition of three or more functions?

Tip: Always check that the range of the inner function aligns with the domain of the outer function when composing functions.