1. **Problem Statement:** Given two functions $f(x) = 2x$ and $g(x) = x + 9$, find: a. $(f \circ g)(x)$ b. $(g \circ f)(x)$ c. $(f \circ g)(3)$ d. $(g \circ f)(3)$ 2. **Formula and Explanation:** - The composition of functions $(f \circ g)(x)$ means $f(g(x))$, which is applying $g$ first, then $f$. - Similarly, $(g \circ f)(x) = g(f(x))$. 3. **Step-by-step Solutions:** a. Find $(f \circ g)(x) = f(g(x))$: - Substitute $g(x)$ into $f$: $f(g(x)) = f(x + 9)$ - Since $f(x) = 2x$, replace $x$ with $x + 9$: $$f(x + 9) = 2(x + 9)$$ - Simplify: $$2x + 18$$ So, $(f \circ g)(x) = 2x + 18$. b. Find $(g \circ f)(x) = g(f(x))$: - Substitute $f(x)$ into $g$: $g(f(x)) = g(2x)$ - Since $g(x) = x + 9$, replace $x$ with $2x$: $$g(2x) = 2x + 9$$ So, $(g \circ f)(x) = 2x + 9$. c. Find $(f \circ g)(3)$: - First find $g(3)$: $$g(3) = 3 + 9 = 12$$ - Then find $f(g(3)) = f(12)$: $$f(12) = 2 \times 12 = 24$$ So, $(f \circ g)(3) = 24$. d. Find $(g \circ f)(3)$: - First find $f(3)$: $$f(3) = 2 \times 3 = 6$$ - Then find $g(f(3)) = g(6)$: $$g(6) = 6 + 9 = 15$$ So, $(g \circ f)(3) = 15$. 4. **Final answers:** - a. $(f \circ g)(x) = 2x + 18$ - b. $(g \circ f)(x) = 2x + 9$ - c. $(f \circ g)(3) = 24$ - d. $(g \circ f)(3) = 15$