https://file.aiquickdraw.com/mathbot/file/1727842918124nif0wku4.jpg

Let's examine the functions \( g \) and \( f \) in the diagram.

We have two functions:
1. \( g: \text{Domain of } g \to \text{Range of } g \)
2. \( f: \text{Domain of } f \to \text{Range of } f \)

The composition of functions \( f \circ g \) means applying \( g \) first, and then applying \( f \) to the result of \( g \). We will identify the domain and range of this composition.

### Function \( g \):
- Domain: \( \{1, 2, 3, 4, 5\} \)
- Range: \( \{5, 7, 8, 9\} \)

### Function \( f \):
- Domain: \( \{5, 7, 8, 9\} \) (corresponds to the range of \( g \))
- Range: \( \{2, 3, 5\} \)

### Finding the domain and range of \( f \circ g \):

#### Domain of \( f \circ g \):
The domain of the composition \( f \circ g \) is the set of all values in the domain of \( g \) that get mapped to a value in the domain of \( f \). Since the range of \( g \) is \( \{5, 7, 8, 9\} \) and all these values are in the domain of \( f \), the domain of \( f \circ g \) is the same as the domain of \( g \).

Thus, the **domain of \( f \circ g \)** is:
\[
\{1, 2, 3, 4, 5\}
\]

#### Range of \( f \circ g \):
To find the range, apply \( f \) to the range of \( g \).
- \( g(1) = 7 \), and \( f(7) = 3 \)
- \( g(2) = 9 \), and \( f(9) = 5 \)
- \( g(3) = 8 \), and \( f(8) = 5 \)
- \( g(4) = 5 \), and \( f(5) = 2 \)
- \( g(5) = 7 \), and \( f(7) = 3 \)

Thus, the **range of \( f \circ g \)** is:
\[
\{2, 3, 5\}
\]

### Final answers:
(a) Domain of \( f \circ g \): \( \{1, 2, 3, 4, 5\} \)

(b) Range of \( f \circ g \): \( \{2, 3, 5\} \)

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Would you like further clarification on any part of the solution? Here are some follow-up questions to deepen understanding:

1. What is the significance of the domain in a function composition?
2. How do we determine if the composition \( f \circ g \) is defined for all inputs from \( g \)?
3. How would the range change if \( f \) were defined differently?
4. Can the domain and range of a function composition ever be equal?
5. How does the concept of function composition apply to real-world problems?

**Tip:** In function composition, always ensure that the range of the first function aligns with the domain of the second function. This guarantees the composition is well-defined!

Find the domain and range of the composition f ∘ g from the given functions g and f, represented in the diagram.

This problem asks to determine the domain and range of the composition f ∘ g given two functions f and g. Explore how to evaluate the domain and range through a detailed step-by-step solution, including checking the values that map between the two functions.

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