1. **Problem Statement:** Find formulas for the compositions of the functions $f(x) = \sqrt{2x + 4}$ and $g(x) = x^{2} + 1$ for the following: i. $f(f(x))$ ii. $f(g(x))$ iii. $g(f(x))$ iv. $g(g(x))$ 2. **Recall the definitions:** - $f(x) = \sqrt{2x + 4}$ means for any input $x$, output is the square root of $2x + 4$. - $g(x) = x^{2} + 1$ means for any input $x$, output is $x$ squared plus 1. 3. **Composition rules:** - $f(f(x))$ means apply $f$ to $x$, then apply $f$ again to the result. - $f(g(x))$ means apply $g$ to $x$, then apply $f$ to that result. - $g(f(x))$ means apply $f$ to $x$, then apply $g$ to that result. - $g(g(x))$ means apply $g$ to $x$, then apply $g$ again to that result. 4. **Calculate each composition:** i. $f(f(x)) = f\big(\sqrt{2x + 4}\big) = \sqrt{2\big(\sqrt{2x + 4}\big) + 4}$ This is simplified as: $$f(f(x)) = \sqrt{2\sqrt{2x + 4} + 4}$$ ii. $f(g(x)) = f(x^{2} + 1) = \sqrt{2(x^{2} + 1) + 4} = \sqrt{2x^{2} + 2 + 4} = \sqrt{2x^{2} + 6}$ iii. $g(f(x)) = g\big(\sqrt{2x + 4}\big) = \big(\sqrt{2x + 4}\big)^{2} + 1 = (2x + 4) + 1 = 2x + 5$ iv. $g(g(x)) = g(x^{2} + 1) = (x^{2} + 1)^{2} + 1 = (x^{2} + 1)(x^{2} + 1) + 1 = x^{4} + 2x^{2} + 1 + 1 = x^{4} + 2x^{2} + 2$ 5. **Summary of results:** - $f(f(x)) = \sqrt{2\sqrt{2x + 4} + 4}$ - $f(g(x)) = \sqrt{2x^{2} + 6}$ - $g(f(x)) = 2x + 5$ - $g(g(x)) = x^{4} + 2x^{2} + 2$ These formulas represent the compositions requested, simplified as much as possible.