Stewart Vega
08/30/2024 · Senior High School
Suppose that the functions \( u \) and \( w \) are defined as follows. \[ \begin{array}{l}u(x)=x^{2}+9 \\ w(x)=\sqrt{x+8} \\ \text { Find the following. } \\ \qquad(w \circ u)(8)=\square \\ (u \circ w)(8)=\square\end{array} \] \[ \]
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find \((w \circ u)(8)\) and \((u \circ w)(8)\), we will first compute \(u(8)\) and then use that result to find \(w(u(8))\). Then we will compute \(w(8)\) and use that result to find \(u(w(8))\).
1. **Calculate \(u(8)\)**:
\[
u(x) = x^2 + 9
\]
\[
u(8) = 8^2 + 9 = 64 + 9 = 73
\]
2. **Calculate \((w \circ u)(8)\)**:
\[
(w \circ u)(8) = w(u(8)) = w(73)
\]
Now we need to calculate \(w(73)\):
\[
w(x) = \sqrt{x + 8}
\]
\[
w(73) = \sqrt{73 + 8} = \sqrt{81} = 9
\]
Thus, \((w \circ u)(8) = 9\).
3. **Calculate \(w(8)\)**:
\[
w(8) = \sqrt{8 + 8} = \sqrt{16} = 4
\]
4. **Calculate \((u \circ w)(8)\)**:
\[
(u \circ w)(8) = u(w(8)) = u(4)
\]
Now we need to calculate \(u(4)\):
\[
u(4) = 4^2 + 9 = 16 + 9 = 25
\]
Thus, \((u \circ w)(8) = 25\).
In summary, we have:
\[
(w \circ u)(8) = 9 \quad \text{and} \quad (u \circ w)(8) = 25
\]
Final answers:
\[
(w \circ u)(8) = 9 \quad \text{and} \quad (u \circ w)(8) = 25
\]
Quick Answer
\( (w \circ u)(8) = 9 \) and \( (u \circ w)(8) = 25 \)
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