1. **State the problem:** Given the functions $f(x) = 4x + 5$, $g(x) = 2x^2$, and $h(x) = 7 - 2x$, find the compositions $fg(x)$, $gh(x)$, $hg(x)$, $fh(x)$, $fgh(x)$, and $hh(x)$ as algebraic expressions. 2. **Recall the composition rule:** For functions $f$ and $g$, the composition $fg(x)$ means $f(g(x))$, i.e., substitute $g(x)$ into $f$. 3. **Calculate $fg(x)$:** $$fg(x) = f(g(x)) = f(2x^2) = 4(2x^2) + 5 = 8x^2 + 5$$ 4. **Calculate $gh(x)$:** $$gh(x) = g(h(x)) = g(7 - 2x) = 2(7 - 2x)^2 = 2(49 - 28x + 4x^2) = 98 - 56x + 8x^2$$ 5. **Calculate $hg(x)$:** $$hg(x) = h(g(x)) = h(2x^2) = 7 - 2(2x^2) = 7 - 4x^2$$ 6. **Calculate $fh(x)$:** $$fh(x) = f(h(x)) = f(7 - 2x) = 4(7 - 2x) + 5 = 28 - 8x + 5 = 33 - 8x$$ 7. **Calculate $fgh(x)$:** $$fgh(x) = f(g(h(x))) = f(2(7 - 2x)^2) = f(2(49 - 28x + 4x^2)) = f(98 - 56x + 8x^2)$$ $$= 4(98 - 56x + 8x^2) + 5 = 392 - 224x + 32x^2 + 5 = 397 - 224x + 32x^2$$ 8. **Calculate $hh(x)$:** $$hh(x) = h(h(x)) = h(7 - 2x) = 7 - 2(7 - 2x) = 7 - 14 + 4x = -7 + 4x$$ **Final answers:** - $fg(x) = 8x^2 + 5$ - $gh(x) = 98 - 56x + 8x^2$ - $hg(x) = 7 - 4x^2$ - $fh(x) = 33 - 8x$ - $fgh(x) = 397 - 224x + 32x^2$ - $hh(x) = -7 + 4x$