Rogers Fernandez
04/27/2023 · Senior High School
Suppose that the functions \( q \) and \( r \) are defined as follows. \[ \begin{array}{l}q(x)=-5 x+2 \\ r(x)=3 x-2 \\ \text { Find the following. } \\ (q \circ q)(5)=\square \\ (q \circ r)(5)=\square\end{array} \]
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Tutor-Verified Answer
Step-by-step Solution
To find the values of \( (q \circ q)(5) \) and \( (q \circ r)(5) \), we need to evaluate the compositions of the functions \( q \) and \( r \).
1. **Calculating \( (q \circ q)(5) \)**:
This means we first apply \( q \) to 5, and then apply \( q \) again to the result.
First, we calculate \( q(5) \):
\[
q(5) = -5(5) + 2 = -25 + 2 = -23
\]
Now, we apply \( q \) to the result \( -23 \):
\[
q(-23) = -5(-23) + 2 = 115 + 2 = 117
\]
Therefore, \( (q \circ q)(5) = 117 \).
2. **Calculating \( (q \circ r)(5) \)**:
This means we first apply \( r \) to 5, and then apply \( q \) to the result.
First, we calculate \( r(5) \):
\[
r(5) = 3(5) - 2 = 15 - 2 = 13
\]
Now, we apply \( q \) to the result \( 13 \):
\[
q(13) = -5(13) + 2 = -65 + 2 = -63
\]
Therefore, \( (q \circ r)(5) = -63 \).
In summary, the results are:
\[
(q \circ q)(5) = 117 \quad \text{and} \quad (q \circ r)(5) = -63
\]
Quick Answer
\( (q \circ q)(5) = 117 \) and \( (q \circ r)(5) = -63 \)
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