Alright, we are given the functions [tex]\( q(x) = x^2 + 7 \)[/tex] and [tex]\( r(x) = \sqrt{x+4} \)[/tex]. We need to find [tex]\((q \circ r)(5)\)[/tex] and [tex]\((r \circ q)(5)\)[/tex].
Let's start with [tex]\((q \circ r)(5)\)[/tex]:
1. First, we evaluate [tex]\( r(5) \)[/tex]:
[tex]\[
r(5) = \sqrt{5 + 4} = \sqrt{9} = 3
\][/tex]
2. Then, we use this result as the input for [tex]\( q \)[/tex]:
[tex]\[
q(r(5)) = q(3)
\][/tex]
Now, we compute [tex]\( q(3) \)[/tex]:
[tex]\[
q(3) = 3^2 + 7 = 9 + 7 = 16
\][/tex]
Thus, [tex]\((q \circ r)(5) = 16\)[/tex].
Next, we find [tex]\((r \circ q)(5)\)[/tex]:
1. First, we evaluate [tex]\( q(5) \)[/tex]:
[tex]\[
q(5) = 5^2 + 7 = 25 + 7 = 32
\][/tex]
2. Then, we use this result as the input for [tex]\( r \)[/tex]:
[tex]\[
r(q(5)) = r(32)
\][/tex]
Now, we compute [tex]\( r(32) \)[/tex]:
[tex]\[
r(32) = \sqrt{32 + 4} = \sqrt{36} = 6
\][/tex]
Thus, [tex]\((r \circ q)(5) = 6\)[/tex].
Hence, the solutions are:
[tex]\[
(q \circ r)(5) = 16
\][/tex]
[tex]\[
(r \circ q)(5) = 6
\][/tex]
