George Chadwick
06/08/2023 · Senior High School
Suppose that the functions \( p \) and \( q \) are defined as follows. \[ \begin{array}{l}p(x)=x+1 \\ q(x)=2 x^{2} \\ \text { Find the following. } \\ (p \circ q)(-5)=\square \\ (q \circ p)(-5)=\square\end{array} \]
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find \((p \circ q)(-5)\) and \((q \circ p)(-5)\), we need to evaluate the compositions of the functions \(p\) and \(q\).
1. **Calculating \((p \circ q)(-5)\)**:
\[
(p \circ q)(-5) = p(q(-5))
\]
First, we need to find \(q(-5)\):
\[
q(-5) = 2(-5)^2 = 2 \cdot 25 = 50
\]
Now, we substitute this result into \(p\):
\[
p(50) = 50 + 1 = 51
\]
Therefore,
\[
(p \circ q)(-5) = 51
\]
2. **Calculating \((q \circ p)(-5)\)**:
\[
(q \circ p)(-5) = q(p(-5))
\]
First, we need to find \(p(-5)\):
\[
p(-5) = -5 + 1 = -4
\]
Now, we substitute this result into \(q\):
\[
q(-4) = 2(-4)^2 = 2 \cdot 16 = 32
\]
Therefore,
\[
(q \circ p)(-5) = 32
\]
In summary, the results are:
\[
(p \circ q)(-5) = 51 \quad \text{and} \quad (q \circ p)(-5) = 32
\]
Quick Answer
\[
(p \circ q)(-5) = 51 \quad \text{and} \quad (q \circ p)(-5) = 32
\]
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