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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 \( f \circ g \)**: \(\{1, 6, 7, 9\}\).

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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.

Find the domain and range of the composition f ∘ g based on the given mappings for functions g and f.

Learn how to determine the domain and range of the composition of two functions, f and g, given specific mappings. This problem involves understanding function composition and analyzing domain and range relationships, suitable for high school students.

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