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The image shows two functions, \( f \) and \( g \), with their domains and ranges visually represented. The problem asks to determine the domain and range of the composition \( g \circ f \), where the functions are composed, meaning \( g(f(x)) \).

### Step-by-step solution:

1. **Function \( f \):**
   - The domain of \( f \): \( \{1, 2, 3, 4\} \)
   - The range of \( f \): \( \{3, 4, 5\} \)
   - The mappings for \( f \):
     - \( f(1) = 4 \)
     - \( f(2) = 3 \)
     - \( f(3) = 5 \)
     - \( f(4) = 5 \)

2. **Function \( g \):**
   - The domain of \( g \): \( \{3, 4, 5\} \)
   - The range of \( g \): \( \{1, 2, 6\} \)
   - The mappings for \( g \):
     - \( g(3) = 1 \)
     - \( g(4) = 2 \)
     - \( g(5) = 6 \)

### Composition \( g \circ f \):

To find the composition \( g \circ f \), you apply \( g \) to the outputs of \( f \). That is, for each \( x \) in the domain of \( f \), calculate \( g(f(x)) \):

- \( g(f(1)) = g(4) = 2 \)
- \( g(f(2)) = g(3) = 1 \)
- \( g(f(3)) = g(5) = 6 \)
- \( g(f(4)) = g(5) = 6 \)

### (a) **Domain of \( g \circ f \)**:
The domain of the composition is the domain of \( f \), which is:
\[
\{1, 2, 3, 4\}
\]

### (b) **Range of \( g \circ f \)**:
The range of the composition consists of the outputs of \( g \circ f \), which are:
\[
\{1, 2, 6\}
\]

Thus, the final answers are:

- **Domain of \( g \circ f \)**: \( \{1, 2, 3, 4\} \)
- **Range of \( g \circ f \)**: \( \{1, 2, 6\} \)

Would you like more details or have any questions about this? Here are some related questions for further practice:

1. How do you determine if a composition of two functions is defined?
2. Can the range of \( f \) ever exceed the domain of \( g \)?
3. What happens if the range of \( f \) has elements outside the domain of \( g \)?
4. How does the composition of functions relate to real-world applications?
5. What are the domain and range of the inverse of a composition?

**Tip:** When composing functions, always check that the output of the first function fits within the domain of the second.

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

Learn how to find the domain and range of the composition of two functions g∘f using set notation. This problem includes the step-by-step breakdown of applying function composition and analyzing the mappings for both functions f and g. Suitable for high school students studying function compositions.

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