In this problem, you are asked to find the domain and range of the composition of functions $g \circ f$, given the mappings of $f$ and $g$ between their respective domains and ranges.

Step 1: Analyze Function $f$

From the left diagram:

• Domain of $f$: {4, 5, 6, 8}
• Range of $f$: {0, 1, 3, 5}

The mappings of $f$ are:

• $f(4) = 0$
• $f(5) = 1$
• $f(6) = 3$
• $f(8) = 5$

Step 2: Analyze Function $g$

From the right diagram:

• Domain of $g$: {0, 1, 3, 5} (which is the same as the range of $f$)
• Range of $g$: {2, 4, 6, 7}

The mappings of $g$ are:

• $g(0) = 4$
• $g(1) = 2$
• $g(3) = 7$
• $g(5) = 6$

Step 3: Composition $g \circ f$

The composition $g \circ f$ means applying $f$ first, and then applying $g$ to the result. Let's go step by step for each element in the domain of $f$:

• $(g \circ f)(4) = g(f(4)) = g(0) = 4$
• $(g \circ f)(5) = g(f(5)) = g(1) = 2$
• $(g \circ f)(6) = g(f(6)) = g(3) = 7$
• $(g \circ f)(8) = g(f(8)) = g(5) = 6$

Step 4: Find Domain and Range of $g \circ f$

• Domain of $g \circ f$: This is the domain of $f$, which is {4, 5, 6, 8}.
• Range of $g \circ f$: The outputs of the composition $g \circ f$ are {4, 2, 7, 6}.

Final Answer:

(a) Domain of $g \circ f$: {4, 5, 6, 8}

(b) Range of $g \circ f$: {2, 4, 6, 7}

Would you like any more details or clarifications on these concepts?

Here are 5 related questions for further practice:

1. How do you find the composition of two functions algebraically, without mappings?
2. What is the domain of a composition of functions if one function is undefined for certain values in its domain?
3. Can the composition of two functions ever result in a function with a different domain than the original functions?
4. How does the range of a function affect the composition with another function?
5. Can functions be composed in the reverse order, i.e., $f \circ g$? How would the process differ?

Tip: Always check if the range of the first function matches the domain of the second function when composing two functions.