Sure, let's solve this step by step.
### Finding [tex]\((r \circ q)(7)\)[/tex]
The notation [tex]\((r \circ q)(7)\)[/tex] means we first apply the function [tex]\(q\)[/tex] to the number 7, and then apply the function [tex]\(r\)[/tex] to the result from [tex]\(q\)[/tex].
1. First, calculate [tex]\(q(7)\)[/tex]:
[tex]\[
q(x) = x^2 + 6
\][/tex]
Substituting [tex]\(x = 7\)[/tex]:
[tex]\[
q(7) = 7^2 + 6 = 49 + 6 = 55
\][/tex]
2. Next, use this result to find [tex]\(r(q(7))\)[/tex] which is [tex]\(r(55)\)[/tex]:
[tex]\[
r(x) = \sqrt{x + 9}
\][/tex]
Substituting [tex]\(x = 55\)[/tex]:
[tex]\[
r(55) = \sqrt{55 + 9} = \sqrt{64} = 8
\][/tex]
Thus, [tex]\((r \circ q)(7) = 8.0\)[/tex].
### Finding [tex]\((q \circ r)(7)\)[/tex]
The notation [tex]\((q \circ r)(7)\)[/tex] means we first apply the function [tex]\(r\)[/tex] to the number 7, and then apply the function [tex]\(q\)[/tex] to the result from [tex]\(r\)[/tex].
1. First, calculate [tex]\(r(7)\)[/tex]:
[tex]\[
r(x) = \sqrt{x + 9}
\][/tex]
Substituting [tex]\(x = 7\)[/tex]:
[tex]\[
r(7) = \sqrt{7 + 9} = \sqrt{16} = 4
\][/tex]
2. Next, use this result to find [tex]\(q(r(7))\)[/tex] which is [tex]\(q(4)\)[/tex]:
[tex]\[
q(x) = x^2 + 6
\][/tex]
Substituting [tex]\(x = 4\)[/tex]:
[tex]\[
q(4) = 4^2 + 6 = 16 + 6 = 22
\][/tex]
Thus, [tex]\((q \circ r)(7) = 22.0\)[/tex].
To summarize:
[tex]\[
(r \circ q)(7) = 8.0
\][/tex]
[tex]\[
(q \circ r)(7) = 22.0
\][/tex]
