1. **State the problem:** We are given two functions: $$q(x) = x^2 + 9$$ $$r(x) = \sqrt{x + 8}$$ We need to find the values of the compositions $$(q \circ r)(8)$$ and $$(r \circ q)(8)$$. 2. **Recall the definition of composition:** $$(q \circ r)(x) = q(r(x))$$ means we first apply $r$ to $x$, then apply $q$ to the result. $$(r \circ q)(x) = r(q(x))$$ means we first apply $q$ to $x$, then apply $r$ to the result. 3. **Calculate $(q \circ r)(8)$:** First, find $r(8)$: $$r(8) = \sqrt{8 + 8} = \sqrt{16} = 4$$ Next, apply $q$ to this result: $$q(4) = 4^2 + 9 = 16 + 9 = 25$$ So, $$(q \circ r)(8) = 25$$. 4. **Calculate $(r \circ q)(8)$:** First, find $q(8)$: $$q(8) = 8^2 + 9 = 64 + 9 = 73$$ Next, apply $r$ to this result: $$r(73) = \sqrt{73 + 8} = \sqrt{81} = 9$$ So, $$(r \circ q)(8) = 9$$. **Final answers:** $$(q \circ r)(8) = 25$$ $$(r \circ q)(8) = 9$$