To find [tex]\((u \circ w)(7)\)[/tex] and [tex]\((w \circ u)(7)\)[/tex], we need to evaluate the composite functions [tex]\(u(w(x))\)[/tex] and [tex]\(w(u(x))\)[/tex] at [tex]\(x = 7\)[/tex].
### Part 1: [tex]\((u \circ w)(7)\)[/tex]
1. Define [tex]\(w(7)\)[/tex]:
[tex]\[ w(x) = \sqrt{x + 9} \][/tex]
So,
[tex]\[ w(7) = \sqrt{7 + 9} = \sqrt{16} = 4 \][/tex]
2. Compute [tex]\(u(w(7))\)[/tex]:
[tex]\[ u(x) = x^2 + 6 \][/tex]
Using the result of [tex]\(w(7)\)[/tex],
[tex]\[ u(w(7)) = u(4) \][/tex]
[tex]\[ u(4) = 4^2 + 6 = 16 + 6 = 22 \][/tex]
Thus,
[tex]\[ (u \circ w)(7) = 22 \][/tex]
### Part 2: [tex]\((w \circ u)(7)\)[/tex]
1. Define [tex]\(u(7)\)[/tex]:
[tex]\[ u(x) = x^2 + 6 \][/tex]
So,
[tex]\[ u(7) = 7^2 + 6 = 49 + 6 = 55 \][/tex]
2. Compute [tex]\(w(u(7))\)[/tex]:
[tex]\[ w(x) = \sqrt{x + 9} \][/tex]
Using the result of [tex]\(u(7)\)[/tex],
[tex]\[ w(u(7)) = w(55) \][/tex]
[tex]\[ w(55) = \sqrt{55 + 9} = \sqrt{64} = 8 \][/tex]
Thus,
[tex]\[ (w \circ u)(7) = 8 \][/tex]
In conclusion:
[tex]\[ (u \circ w)(7) = 22 \][/tex]
[tex]\[ (w \circ u)(7) = 8 \][/tex]
