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 $f \circ g$: $\{2, 3, 5\}$

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!