1. **State the problem:** We are given three functions: $f(x) = 3x + 2$, $g(x) = 7 - 2x$, and $h(x) = x - 5$. We need to find and simplify the composite functions $f \circ g (x)$ and $h \circ f (x)$. 2. **Recall the formula for composite functions:** The composite function $f \circ g (x)$ means $f(g(x))$, which is the function $f$ applied to the output of $g(x)$. Similarly, $h \circ f (x) = h(f(x))$. 3. **Find $f \circ g (x)$:** Start with $g(x) = 7 - 2x$. Substitute $g(x)$ into $f$: $$f(g(x)) = 3(7 - 2x) + 2$$ Distribute the 3: $$f(g(x)) = 21 - 6x + 2$$ Combine like terms: $$f(g(x)) = 23 - 6x$$ 4. **Find $h \circ f (x)$:** Start with $f(x) = 3x + 2$. Substitute $f(x)$ into $h$: $$h(f(x)) = (3x + 2) - 5$$ Simplify: $$h(f(x)) = 3x - 3$$ **Final answers:** $$f \circ g (x) = 23 - 6x$$ $$h \circ f (x) = 3x - 3$$