1. **State the problem:** We have three functions: $f(x) = 4x + 5$, $g(x) = 2x^2$, and $h(x) = 7 - 2x$. We want to find the compositions $hg(x)$, $fh(x)$, $fgh(x)$, and $hh(x)$. 2. **Recall function composition:** For functions $a(x)$ and $b(x)$, the composition $a(b(x))$ means substitute $b(x)$ into $a$. 3. **Calculate $hg(x)$:** This means $h(g(x))$. Substitute $g(x) = 2x^2$ into $h(x) = 7 - 2x$. $$hg(x) = h(g(x)) = 7 - 2(2x^2) = 7 - 4x^2$$ 4. **Calculate $fh(x)$:** This means $f(h(x))$. Substitute $h(x) = 7 - 2x$ into $f(x) = 4x + 5$. $$fh(x) = f(h(x)) = 4(7 - 2x) + 5 = 28 - 8x + 5 = 33 - 8x$$ 5. **Calculate $fgh(x)$:** This means $f(g(h(x)))$. First find $h(x)$, then $g(h(x))$, then $f(g(h(x)))$. - $h(x) = 7 - 2x$ - $g(h(x)) = 2(7 - 2x)^2$ Expand $(7 - 2x)^2 = 49 - 28x + 4x^2$. So $$g(h(x)) = 2(49 - 28x + 4x^2) = 98 - 56x + 8x^2$$ Now substitute into $f$: $$f(g(h(x))) = 4(98 - 56x + 8x^2) + 5 = 392 - 224x + 32x^2 + 5 = 397 - 224x + 32x^2$$ 6. **Calculate $hh(x)$:** This means $h(h(x))$. Substitute $h(x) = 7 - 2x$ into $h$. $$hh(x) = h(h(x)) = 7 - 2(7 - 2x) = 7 - 14 + 4x = -7 + 4x$$ **Final answers:** - $hg(x) = 7 - 4x^2$ - $fh(x) = 33 - 8x$ - $fgh(x) = 397 - 224x + 32x^2$ - $hh(x) = -7 + 4x$