To find the domain and range of the composition $f \circ g$ (where $f \circ g(x) = f(g(x))$), we need to analyze both functions $g$ and $f$ based on the mappings shown in the diagrams.

Step 1: Analyze $g$

1. Domain of $g$: $\{0, 1, 3, 5, 8\}$.
2. Range of $g$ (i.e., outputs of $g$ for the inputs in its domain): $\{0, 2, 4, 5, 6\}$.

Step 2: Analyze $f$

1. Domain of $f$: $\{0, 3, 4, 5, 8, 9\}$.
2. Range of $f$ (i.e., outputs of $f$ for the inputs in its domain): $\{1, 4, 6, 7, 9\}$.

Step 3: Determine $f \circ g$

For the composition $f \circ g$ to be defined at a point $x$, $g(x)$ must be in the domain of $f$.

• $g(0) = 0$, and $f(0) = 1$.
• $g(1) = 2$, but 2 is not in the domain of $f$, so $f(g(1))$ is undefined.
• $g(3) = 4$, and $f(4) = 6$.
• $g(5) = 5$, and $f(5) = 7$.
• $g(8) = 6$, and $f(6) = 9$.

From this, we see that $f \circ g$ is defined for inputs $\{0, 3, 5, 8\}$.

Final Answer:

• Domain of $f \circ g$: $\{0, 3, 5, 8\}$.
• Range of $f \circ g$: $\{1, 6, 7, 9\}$.

Would you like a detailed explanation of any step?

Here are some related questions you might consider:

1. How do we determine the composition of two functions in general?
2. What if the function $g$ had additional elements in its domain or range?
3. How would the composition $g \circ f$ differ from $f \circ g$?
4. What are some practical applications of function compositions?
5. How would an undefined composition value impact the domain?

Tip: When working with compositions, always check if the output of the first function (range) falls within the domain of the second function. This ensures the composition is well-defined.