Question

Supposed that the functions Q and R defined as follows.

Answer 1

`\begin{gathered} (r\circ q)(7)=2\sqrt[]{21} \\ (q\circ r)(7)=22 \end{gathered}`

Explanation:

To evaluate composite functions we first evaluate the ''inner'' function, then we evaluate the ''outer'' function. For the following functions:

`\begin{gathered} q(x)=x^2+6 \\ r(x)=\sqrt[]{x+9} \\ \text{ Then, for q}\circ r\colon \\ (q\circ r)(x)=r(q(x)) \\ q(r(x))=(\sqrt[]{x+9})^2+6 \\ q(r(x))=x+9+6 \\ q(r(x))=x+15 \\ \text{ Substitute x=7 in q}(r(x)) \\ q(r(7))=7+15 \\ q(r(7))=22 \end{gathered}`

Now, for the composition of (r o q)(x):

`\begin{gathered} (r\circ q)(x)=r(q(x)) \\ r(q(x))=\sqrt[]{(x^2+6)+9} \\ r(q(x))=\sqrt[]{x^2+15} \\ \text{ Substitute x=7 in r(q(x))} \\ r(q(7))=\sqrt[]{(7)^2+15} \\ r(q(7))=\sqrt[]{64}=2\sqrt[]{21} \end{gathered}`