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To find the domain and range of the composition \( f \circ g \), we first need to understand how \( f \) and \( g \) are mapped.

### **1. Analysis of Functions \( g \) and \( f \):**
- **Function \( g \)** maps:
  - \( 3 \rightarrow 2 \)
  - \( 4 \rightarrow 1 \)
  - \( 5 \rightarrow 8 \)

- **Function \( f \)** maps:
  - \( 1 \rightarrow 1 \)
  - \( 2 \rightarrow 0 \)
  - \( 8 \rightarrow 7 \)

### **2. Composition \( f \circ g \):**
To find \( f(g(x)) \), we apply \( g \) first, then apply \( f \) to the result of \( g(x) \).

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

### **3. Domain and Range of \( f \circ g \):**
- **Domain** of \( f \circ g \): The domain consists of the elements from the domain of \( g \) that map to elements within the domain of \( f \). Since all elements in the domain of \( g \) (3, 4, 5) map to elements within the domain of \( f \) (1, 2, 8), the domain of \( f \circ g \) is:
  \[
  \{3, 4, 5\}
  \]

- **Range** of \( f \circ g \): This is the set of all values \( f(g(x)) \) as \( x \) varies over the domain of \( g \). From our composition, the range is:
  \[
  \{0, 1, 7\}
  \]

### **Final Answers:**
- **(a) Domain of \( f \circ g \):** \(\{3, 4, 5\}\)
- **(b) Range of \( f \circ g \):** \(\{0, 1, 7\}\)

Would you like more details or have any questions about this solution?

### Related Questions:
1. How do you determine the domain and range of a single function from a mapping diagram?
2. What is the composition of two functions and how is it calculated?
3. Can the range of a composite function include values outside the range of the inner function?
4. What happens to the domain and range of \( f \circ g \) if \( g \) or \( f \) is not defined for certain values?
5. How do you find the inverse of a composite function?

**Tip:** Always verify that the output of the inner function falls within the domain of the outer function when computing compositions.

Find the domain and range of the composition f ∘ g based on the mapping diagrams of functions g and f.

Learn how to find the domain and range of the composition of two functions f and g using mapping diagrams. This example demonstrates how to apply function composition and analyze the domain and range step-by-step. Suitable for College Algebra students.

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