Sure, let's solve for [tex]\((w \circ u)(8)\)[/tex] and [tex]\((u \circ w)(8)\)[/tex] step by step.
### Step-by-Step Solution
Given the functions:
[tex]\[ u(x) = x^2 + 9 \][/tex]
[tex]\[ w(x) = \sqrt{x + 8} \][/tex]
We need to find:
[tex]\[ (w \circ u)(8) \][/tex]
[tex]\[ (u \circ w)(8) \][/tex]
#### 1. Calculate [tex]\( (w \circ u)(8) \)[/tex]
This means we need to find [tex]\( w(u(8)) \)[/tex].
1.1 Find [tex]\( u(8) \)[/tex].
[tex]\[ u(x) = x^2 + 9 \][/tex]
So,
[tex]\[ u(8) = 8^2 + 9 \][/tex]
[tex]\[ u(8) = 64 + 9 \][/tex]
[tex]\[ u(8) = 73 \][/tex]
1.2 Now, find [tex]\( w(u(8)) \)[/tex].
[tex]\[ w(x) = \sqrt{x + 8} \][/tex]
So,
[tex]\[ w(u(8)) = w(73) \][/tex]
[tex]\[ w(73) = \sqrt{73 + 8} \][/tex]
[tex]\[ w(73) = \sqrt{81} \][/tex]
[tex]\[ w(73) = 9 \][/tex]
Therefore,
[tex]\[ (w \circ u)(8) = 9 \][/tex]
#### 2. Calculate [tex]\( (u \circ w)(8) \)[/tex]
This means we need to find [tex]\( u(w(8)) \)[/tex].
2.1 Find [tex]\( w(8) \)[/tex].
[tex]\[ w(x) = \sqrt{x + 8} \][/tex]
So,
[tex]\[ w(8) = \sqrt{8 + 8} \][/tex]
[tex]\[ w(8) = \sqrt{16} \][/tex]
[tex]\[ w(8) = 4 \][/tex]
2.2 Now, find [tex]\( u(w(8)) \)[/tex].
[tex]\[ u(x) = x^2 + 9 \][/tex]
So,
[tex]\[ u(w(8)) = u(4) \][/tex]
[tex]\[ u(4) = 4^2 + 9 \][/tex]
[tex]\[ u(4) = 16 + 9 \][/tex]
[tex]\[ u(4) = 25 \][/tex]
Therefore,
[tex]\[ (u \circ w)(8) = 25 \][/tex]
### Summary of Results
[tex]\[ (w \circ u)(8) = 9 \][/tex]
[tex]\[ (u \circ w)(8) = 25 \][/tex]
