Certainly! Let us solve for [tex]\((u \circ w)(1)\)[/tex] and [tex]\((w \circ u)(1)\)[/tex] step by step:
### Finding [tex]\((u \circ w)(1)\)[/tex]:
1. Evaluate [tex]\(w(1)\)[/tex]:
[tex]\( w(x) = \sqrt{x + 3} \)[/tex]
[tex]\[
w(1) = \sqrt{1 + 3} = \sqrt{4} = 2
\][/tex]
2. Substitute [tex]\( w(1) \)[/tex] into [tex]\( u \)[/tex]:
[tex]\( u(x) = x^2 + 5 \)[/tex]
[tex]\[
u(w(1)) = u(2)
\][/tex]
[tex]\[
u(2) = 2^2 + 5 = 4 + 5 = 9
\][/tex]
So, [tex]\((u \circ w)(1) = 9\)[/tex].
### Finding [tex]\((w \circ u)(1)\)[/tex]:
1. Evaluate [tex]\(u(1)\)[/tex]:
[tex]\( u(x) = x^2 + 5 \)[/tex]
[tex]\[
u(1) = 1^2 + 5 = 1 + 5 = 6
\][/tex]
2. Substitute [tex]\( u(1) \)[/tex] into [tex]\( w \)[/tex]:
[tex]\( w(x) = \sqrt{x + 3} \)[/tex]
[tex]\[
w(u(1)) = w(6)
\][/tex]
[tex]\[
w(6) = \sqrt{6 + 3} = \sqrt{9} = 3
\][/tex]
So, [tex]\((w \circ u)(1) = 3\)[/tex].
Therefore, the required values are:
[tex]\[
(u \circ w)(1) = 9
\][/tex]
[tex]\[
(w \circ u)(1) = 3
\][/tex]
