https://file.aiquickdraw.com/mathbot/file/1727301101107u9utf8iv.jpg

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.

Find the domain and range of the composition f ∘ g using the given mappings of two functions g and f.

Learn how to find the domain and range of the composition of two functions f ∘ g using given mappings. Understand key concepts of function composition, and analyze the domain and range step-by-step. Suitable for high school students in Grades 10-12.

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