1. **Problem statement:** Find the domain and range of the composition $g \circ f$, where $f$ and $g$ are functions with given domains and ranges. 2. **Recall the composition definition:** The composition $g \circ f$ means applying $f$ first, then applying $g$ to the result: $$(g \circ f)(x) = g(f(x))$$ 3. **Domain of $g \circ f$:** The domain of $g \circ f$ consists of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$. 4. **Given data:** - Domain of $f = \{2,3,4,5,9\}$ - Range of $f = \{1,3,5,7,9\}$ - Domain of $g = \{1,5,6,7,9\}$ - Range of $g = \{5,7,8,9\}$ 5. **Check which $f(x)$ values are in domain of $g$:** Range of $f$ is $\{1,3,5,7,9\}$ Domain of $g$ is $\{1,5,6,7,9\}$ The values $1,5,7,9$ are in domain of $g$, but $3$ is not. 6. **Find domain of $g \circ f$ by selecting $x$ in domain of $f$ where $f(x)$ is in domain of $g$:** From the function $f$: - $f(2) = 1$ (in domain of $g$) - $f(3) = 3$ (not in domain of $g$) - $f(4) = 5$ (in domain of $g$) - $f(5) = 7$ (in domain of $g$) - $f(9) = 9$ (in domain of $g$) So domain of $g \circ f = \{2,4,5,9\}$ 7. **Find range of $g \circ f$ by applying $g$ to $f(x)$ for $x$ in domain of $g \circ f$:** - $(g \circ f)(2) = g(f(2)) = g(1) = 5$ - $(g \circ f)(4) = g(f(4)) = g(5) = 7$ - $(g \circ f)(5) = g(f(5)) = g(7) = 9$ - $(g \circ f)(9) = g(f(9)) = g(9) = 9$ Range of $g \circ f = \{5,7,9\}$ **Final answers:** (a) Domain of $g \circ f = \{2,4,5,9\}$ (b) Range of $g \circ f = \{5,7,9\}$