1. **State the problem:** We have two functions $f$ and $g$ with given domains and ranges. We want to find the domain and range of the composition $g \circ f$, which means $g(f(x))$. 2. **Recall the definition of composition:** The domain of $g \circ f$ consists of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$. The range of $g \circ f$ is the set of all values $g(f(x))$ for $x$ in the domain of $g \circ f$. 3. **Given data:** - Domain of $f = \{1, 2, 5, 7\}$ - Range of $f = \{0, 6, 8\}$ with mappings: $1 \to 0$, $2 \to 6$, $5 \to 8$, $7 \to 0$ - Domain of $g = \{0, 1, 2, 3, 6, 9\}$ - Range of $g = \{0, 4\}$ with mappings: $0 \to 4$, $1 \to 0$, $2 \to 0$, $3 \to 0$, $6 \to 4$, $9 \to 0$ 4. **Find domain of $g \circ f$:** We check which $x$ in domain of $f$ have $f(x)$ in domain of $g$. - $f(1) = 0$, and $0 \in$ domain of $g$ (yes) - $f(2) = 6$, and $6 \in$ domain of $g$ (yes) - $f(5) = 8$, but $8 \notin$ domain of $g$ (no) - $f(7) = 0$, and $0 \in$ domain of $g$ (yes) So domain of $g \circ f = \{1, 2, 7\}$. 5. **Find range of $g \circ f$:** Calculate $g(f(x))$ for $x$ in domain of $g \circ f$: - $g(f(1)) = g(0) = 4$ - $g(f(2)) = g(6) = 4$ - $g(f(7)) = g(0) = 4$ Range of $g \circ f = \{4\}$. **Final answers:** (a) Domain of $g \circ f = \{1, 2, 7\}$ (b) Range of $g \circ f = \{4\}$