To solve this, let's analyze the composition of functions $f \circ g$ based on the mappings provided.

Given Information:

1. The function $g$ maps elements from its domain to its range.

   • Domain of $g$: $\{1, 3, 6, 7, 8, 9\}$
   • Range of $g$: $\{2, 5, 7\}$

   Mapping for $g$:

   • $g(1) = 2$
   • $g(3) = 5$
   • $g(6) = 5$
   • $g(7) = 7$
   • $g(8) = 2$
   • $g(9) = 7$

2. The function $f$ maps elements from its domain to its range.

   • Domain of $f$: $\{2, 4, 5, 6, 7, 8\}$
   • Range of $f$: $\{0, 5, 6, 8\}$

   Mapping for $f$:

   • $f(2) = 0$
   • $f(4) = 6$
   • $f(5) = 5$
   • $f(6) = 8$
   • $f(7) = 6$
   • $f(8) = 8$

(a) Domain of $f \circ g$:

The domain of $f \circ g$ includes elements from the domain of $g$ that map to elements within the domain of $f$. To determine this, we check each output of $g$ and ensure it's in the domain of $f$:

• $g(1) = 2$, which is in the domain of $f$.
• $g(3) = 5$, which is in the domain of $f$.
• $g(6) = 5$, which is in the domain of $f$.
• $g(7) = 7$, which is in the domain of $f$.
• $g(8) = 2$, which is in the domain of $f$.
• $g(9) = 7$, which is in the domain of $f$.

Thus, all elements in the domain of $g$ map to values in the domain of $f$, so the domain of $f \circ g$ is: $\{1, 3, 6, 7, 8, 9\}$

(b) Range of $f \circ g$:

To find the range of $f \circ g$, we apply $f$ to each value that $g$ maps to:

• $f(g(1)) = f(2) = 0$
• $f(g(3)) = f(5) = 5$
• $f(g(6)) = f(5) = 5$
• $f(g(7)) = f(7) = 6$
• $f(g(8)) = f(2) = 0$
• $f(g(9)) = f(7) = 6$

The unique values obtained are $0$, $5$, and $6$.

Therefore, the range of $f \circ g$ is: $\{0, 5, 6\}$

Final Answers:

• (a) Domain of $f \circ g$: $\{1, 3, 6, 7, 8, 9\}$
• (b) Range of $f \circ g$: $\{0, 5, 6\}$

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. How does the domain of a composite function depend on the individual functions?
2. Why is it necessary for the range of $g$ to be a subset of the domain of $f$ in the composition $f \circ g$?
3. What would happen if $g$ mapped to values outside the domain of $f$?
4. Can you find an example where the range of $f \circ g$ is identical to the range of $f$?
5. How can we determine the range of $f \circ g$ if the mappings are not explicitly provided?

Tip: When dealing with compositions, always confirm that each function's output can be used as the next function's input. This is key to finding the domain of the composite function.