Question

q(x)=x²+5
r(x)=√(x+7)
find the following.
(r∘q)(2) =
(q∘r)(2) =

Explanation:

Step1: Find \((r \circ q)(2)\)

First, recall that \((r \circ q)(x)=r(q(x))\). So we need to find \(q(2)\) first, then substitute it into \(r(x)\).

Find \(q(2)\): Given \(q(x)=x^{2}+5\), substitute \(x = 2\):
\(q(2)=2^{2}+5=4 + 5=9\)

Now substitute \(q(2) = 9\) into \(r(x)\): Given \(r(x)=\sqrt{x + 7}\), so \(r(q(2))=r(9)=\sqrt{9+7}=\sqrt{16}=4\)

Step2: Find \((q \circ r)(2)\)

Recall that \((q \circ r)(x)=q(r(x))\). So we need to find \(r(2)\) first, then substitute it into \(q(x)\).

Find \(r(2)\): Given \(r(x)=\sqrt{x + 7}\), substitute \(x=2\):
\(r(2)=\sqrt{2 + 7}=\sqrt{9}=3\)

Now substitute \(r(2)=3\) into \(q(x)\): Given \(q(x)=x^{2}+5\), so \(q(r(2))=q(3)=3^{2}+5=9 + 5=14\)

Answer:

\((r \circ q)(2)=\boldsymbol{4}\)

\((q \circ r)(2)=\boldsymbol{14}\)