The problem asks us to find the compositions $f(g(x))$ and $g(f(x))$ given $f(x) = \sqrt{-15-x}$ and $g(x) = x^2 - 8x$.

1. Problem Description

The problem asks us to find the compositions $f(g(x))$ and $g(f(x))$ given $f(x) = \sqrt{-15-x}$ and $g(x) = x^2 - 8x$.

2. Solution Steps

First, we find $f(g(x))$. This means we need to substitute $g(x)$ into $f(x)$ wherever we see $x$.

$f(g(x)) = f(x^2 - 8x) = \sqrt{-15 - (x^2 - 8x)} = \sqrt{-15 - x^2 + 8x}$.

Next, we find $g(f(x))$. This means we need to substitute $f(x)$ into $g(x)$ wherever we see $x$.

$g(f(x)) = g(\sqrt{-15-x}) = (\sqrt{-15-x})^2 - 8(\sqrt{-15-x}) = (-15-x) - 8\sqrt{-15-x}$.

$g(f(x)) = -15 - x - 8\sqrt{-15-x}$.

3. Final Answer

$f(g(x)) = \sqrt{-15 - x^2 + 8x}$

$g(f(x)) = -15 - x - 8\sqrt{-15-x}$