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In this problem, you are asked to find the domain and range of the composition of functions \( g \circ f \), given the mappings of \( f \) and \( g \) between their respective domains and ranges.

### Step 1: Analyze Function \( f \)
From the left diagram:
- **Domain of \( f \)**: \{4, 5, 6, 8\}
- **Range of \( f \)**: \{0, 1, 3, 5\}

The mappings of \( f \) are:
- \( f(4) = 0 \)
- \( f(5) = 1 \)
- \( f(6) = 3 \)
- \( f(8) = 5 \)

### Step 2: Analyze Function \( g \)
From the right diagram:
- **Domain of \( g \)**: \{0, 1, 3, 5\} (which is the same as the range of \( f \))
- **Range of \( g \)**: \{2, 4, 6, 7\}

The mappings of \( g \) are:
- \( g(0) = 4 \)
- \( g(1) = 2 \)
- \( g(3) = 7 \)
- \( g(5) = 6 \)

### Step 3: Composition \( g \circ f \)
The composition \( g \circ f \) means applying \( f \) first, and then applying \( g \) to the result. Let's go step by step for each element in the domain of \( f \):

- \( (g \circ f)(4) = g(f(4)) = g(0) = 4 \)
- \( (g \circ f)(5) = g(f(5)) = g(1) = 2 \)
- \( (g \circ f)(6) = g(f(6)) = g(3) = 7 \)
- \( (g \circ f)(8) = g(f(8)) = g(5) = 6 \)

### Step 4: Find Domain and Range of \( g \circ f \)
- **Domain of \( g \circ f \)**: This is the domain of \( f \), which is \{4, 5, 6, 8\}.
- **Range of \( g \circ f \)**: The outputs of the composition \( g \circ f \) are \{4, 2, 7, 6\}.

### Final Answer:
(a) **Domain of \( g \circ f \)**: \{4, 5, 6, 8\}

(b) **Range of \( g \circ f \)**: \{2, 4, 6, 7\}

Would you like any more details or clarifications on these concepts?

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Here are 5 related questions for further practice:
1. How do you find the composition of two functions algebraically, without mappings?
2. What is the domain of a composition of functions if one function is undefined for certain values in its domain?
3. Can the composition of two functions ever result in a function with a different domain than the original functions?
4. How does the range of a function affect the composition with another function?
5. Can functions be composed in the reverse order, i.e., \( f \circ g \)? How would the process differ?

**Tip:** Always check if the range of the first function matches the domain of the second function when composing two functions.

Find the domain and range of the composition g ∘ f. Write the answers in set notation.

Learn how to find the domain and range of the composition of two functions, g ∘ f, with a step-by-step solution. This problem demonstrates function composition, set notation, and is suitable for students in Grades 10-12.

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