1. **State the problem:** We are given two functions $g$ and $f$ with their domains and ranges, and we need to find the domain and range of the composition $f \circ g$. 2. **Recall the definition of composition:** The composition $f \circ g$ means applying $g$ first, then applying $f$ to the result. Formally, $$(f \circ g)(x) = f(g(x)).$$ 3. **Domain of $f \circ g$:** The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$. 4. **Given data:** - Domain of $g = \{0, 2, 4, 5, 8, 9\}$ - Range of $g = \{2, 6, 7, 8\}$ - Domain of $f = \{4, 6, 7, 8\}$ - Range of $f = \{6, 7\}$ 5. **Check which $g(x)$ values are in domain of $f$:** - $g(0) = 2$ (2 not in domain of $f$) - $g(2) = 6$ (6 in domain of $f$) - $g(4) = 6$ (6 in domain of $f$) - $g(5) = 7$ (7 in domain of $f$) - $g(8) = 8$ (8 in domain of $f$) - $g(9) = 7$ (7 in domain of $f$) 6. **Domain of $f \circ g$ is all $x$ in domain of $g$ where $g(x)$ in domain of $f$:** $$\{2, 4, 5, 8, 9\}$$ 7. **Find the range of $f \circ g$ by evaluating $f(g(x))$ for $x$ in domain of $f \circ g$:** - $f(g(2)) = f(6) = 6$ - $f(g(4)) = f(6) = 6$ - $f(g(5)) = f(7) = 7$ - $f(g(8)) = f(8) = 7$ - $f(g(9)) = f(7) = 7$ 8. **Range of $f \circ g$ is the set of these values:** $$\{6, 7\}$$ **Final answers:** (a) Domain of $f \circ g = \{2, 4, 5, 8, 9\}$ (b) Range of $f \circ g = \{6, 7\}$