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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?

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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.

Find the domain and range of the composition f ∘ g based on the mappings provided.

This problem involves determining the domain and range of the composite function f ∘ g using provided mappings for functions f and g. Learn how to evaluate the compatibility of domains and ranges for composition, with a detailed step-by-step solution suitable for Grades 10-12.

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