Suppose that the functions \( q \) and \( r \) are defined as follows \[ \begin{array}{l}q(x)=x^{2}+5 \\ r(x)=\sqrt{x+7} \\ \text { Find the following. }\end{array} \text {. } \]

We are given   q(x) = x² + 5   r(x) = √(x + 7) A common task with two functions defined in this way is to form their composite functions. Two natural composite functions to consider are   (r ∘ q)(x) = r(q(x)) and (q ∘ r)(x) = q(r(x)). Below are the steps to find each of these, along with their domains. ───────────────────────────── Step 1. Find (r ∘ q)(x) By definition, (r ∘ q)(x) means we first apply q to x and then apply r to the result. Thus:   r(q(x)) = √(q(x) + 7) Since q(x) = x² + 5, substitute this in:   r(q(x)) = √(x² + 5 + 7) = √(x² + 12). Domain Consideration for r(q(x)): Recall that the square root function requires its input to be nonnegative. Here, the radicand is   x² + 12. Since x² is always nonnegative for all real x, we have   x² + 12 ≥ 12 > 0 for all x. Thus, (r ∘ q)(x) = √(x² + 12) is defined for all real numbers. ───────────────────────────── Step 2. Find (q ∘ r)(x) Now, (q ∘ r)(x) means we first apply r to x and then apply q to the result. Thus:   q(r(x)) = (r(x))² + 5. Since r(x) = √(x + 7), substitute this:   q(r(x)) = (√(x + 7))² + 5 = (x + 7) + 5 = x + 12. Domain Consideration for q(r(x)): For r(x) = √(x + 7) to be defined, the radicand must satisfy   x + 7 ≥ 0 ⟹ x ≥ −7. So the composite function (q ∘ r)(x) = x + 12 is defined only for x ≥ −7. ───────────────────────────── Summary of Answers 1. (r ∘ q)(x) = √(x² + 12)   Domain: All real numbers. 2. (q ∘ r)(x) = x + 12      Domain: x ≥ −7. These are the answers for the composite functions formed from q(x) and r(x).