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We are asked to find the domain and range of the composition of functions \( g \circ f \), given the mappings of the two functions \( f \) and \( g \) from the diagrams.

### Step 1: Understanding the function mappings
- Function \( f \) takes inputs from its domain (leftmost oval) and maps them to its range (middle oval).
  - From the diagram, the function \( f \) maps:
    - \( f(0) = 1 \)
    - \( f(2) = 7 \)
    - \( f(4) = 2 \)
    - \( f(6) = 8 \)
    - \( f(8) = 2 \)
    - \( f(9) = 8 \)

- Function \( g \) takes inputs from its domain (middle oval) and maps them to its range (rightmost oval).
  - From the diagram, the function \( g \) maps:
    - \( g(1) = 2 \)
    - \( g(2) = 6 \)
    - \( g(7) = 9 \)
    - \( g(8) = 7 \)

### Step 2: Composing the functions \( g \circ f \)
The composition \( g \circ f \) means \( g(f(x)) \). We substitute the output of \( f(x) \) into \( g \). Let's compute \( g(f(x)) \) for each \( x \) in the domain of \( f \):

- For \( x = 0 \), \( f(0) = 1 \), and \( g(1) = 2 \). So, \( g(f(0)) = 2 \).
- For \( x = 2 \), \( f(2) = 7 \), and \( g(7) = 9 \). So, \( g(f(2)) = 9 \).
- For \( x = 4 \), \( f(4) = 2 \), and \( g(2) = 6 \). So, \( g(f(4)) = 6 \).
- For \( x = 6 \), \( f(6) = 8 \), and \( g(8) = 7 \). So, \( g(f(6)) = 7 \).
- For \( x = 8 \), \( f(8) = 2 \), and \( g(2) = 6 \). So, \( g(f(8)) = 6 \).
- For \( x = 9 \), \( f(9) = 8 \), and \( g(8) = 7 \). So, \( g(f(9)) = 7 \).

### Step 3: Determining the domain and range of \( g \circ f \)

- The **domain** of \( g \circ f \) consists of all values \( x \) in the domain of \( f \) that can be mapped through the composition:
  - Domain of \( f \) is \( \{0, 2, 4, 6, 8, 9\} \).
  
- The **range** of \( g \circ f \) is the set of all possible outputs from \( g(f(x)) \):
  - From the calculations, the outputs are \( \{2, 9, 6, 7\} \).

### Final Answer:
- **Domain of \( g \circ f \)**: \( \{0, 2, 4, 6, 8, 9\} \)
- **Range of \( g \circ f \)**: \( \{2, 6, 7, 9\} \)

Would you like any clarifications or further details? Here are five related questions to think about:

1. What happens if a value from the domain of \( f \) does not have a corresponding mapping in \( g \)?
2. How would the composition change if one of the mappings in \( f \) or \( g \) was different?
3. Can a function have multiple outputs for the same input? Why or why not?
4. What are the conditions for the composition \( g \circ f \) to be defined?
5. How would you determine the composition if \( f \) or \( g \) was not explicitly shown as a diagram?

**Tip:** When working with function compositions, ensure the range of the first function aligns with the domain of the second function for the composition to be valid.

Find the domain and range of the composition g ∘ f from the diagram of functions f and g.

Learn how to find the domain and range of the composition of functions g ∘ f using function mappings. Includes step-by-step instructions and example mappings.

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