1. **Problem Statement:** Find the domain and range of the composition $f \circ g$, where $f$ and $g$ are functions with given domains and ranges. 2. **Recall:** The composition $f \circ g$ means applying $g$ first, then $f$ to the result: $$(f \circ g)(x) = f(g(x)).$$ 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:** - Domain of $g = \{0,1,4,6,7\}$ - Range of $g = \{1,2,3,4,6\}$ - Domain of $f = \{1,2,3,4,7,8\}$ 5. **Check which $g(x)$ values are in domain of $f$:** - $g(0) = 1$, and $1 \in$ domain of $f$ (valid) - $g(1) = 2$, and $2 \in$ domain of $f$ (valid) - $g(4) = 3$, and $3 \in$ domain of $f$ (valid) - $g(6) = 4$, and $4 \in$ domain of $f$ (valid) - $g(7) = 6$, but $6 \notin$ domain of $f$ (invalid) 6. **Therefore, domain of $f \circ g$ is:** $$\{0,1,4,6\}$$ 7. **Range of $f \circ g$:** Apply $f$ to the values $g(x)$ for $x$ in the domain of $f \circ g$: - $f(g(0)) = f(1) = 1$ - $f(g(1)) = f(2) = 3$ - $f(g(4)) = f(3) = 4$ - $f(g(6)) = f(4) = 6$ 8. **Range of $f \circ g$ is:** $$\{1,3,4,6\}$$ **Final answers:** (a) Domain of $f \circ g = \{0,1,4,6\}$ (b) Range of $f \circ g = \{1,3,4,6\}$