Question
Suppose that the functions \( q \) and \( r \) \[ \begin{array}{l}q(x)=x-2 \\ r(x)=-x^{2}-1 \\ \text { Find the following. } \\ (r \circ q)(-4)=\square \\ (q \circ r)(-4)=\square\end{array} \]
Ask by Davison Bolton. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (r \circ q)(-4) = -37 \) and \( (q \circ r)(-4) = -19 \).
Solution
1. Compute \( (r \circ q)(-4) \):
- First, evaluate \( q(-4) \):
\[
q(-4) = -4 - 2 = -6
\]
- Next, evaluate \( r(-6) \):
\[
r(-6) = -(-6)^2 - 1 = -(36) - 1 = -37
\]
- Thus, \( (r \circ q)(-4) = -37 \).
2. Compute \( (q \circ r)(-4) \):
- First, evaluate \( r(-4) \):
\[
r(-4) = -(-4)^2 - 1 = -(16) - 1 = -17
\]
- Next, evaluate \( q(-17) \):
\[
q(-17) = -17 - 2 = -19
\]
- Thus, \( (q \circ r)(-4) = -19 \).
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To solve for \( (r \circ q)(-4) \) and \( (q \circ r)(-4) \), we start by evaluating each function step by step. First, we compute \( (r \circ q)(-4) \): 1. Calculate \( q(-4) \): \[ q(-4) = -4 - 2 = -6 \] 2. Now input this result into \( r(x) \): \[ r(-6) = -(-6)^2 - 1 = -36 - 1 = -37 \] Hence, \( (r \circ q)(-4) = -37 \). Now we compute \( (q \circ r)(-4) \): 1. Calculate \( r(-4) \): \[ r(-4) = -(-4)^2 - 1 = -16 - 1 = -17 \] 2. Now input this result into \( q(x) \): \[ q(-17) = -17 - 2 = -19 \] Thus, \( (q \circ r)(-4) = -19 \). Putting it all together, we have: \[ (r \circ q)(-4) = -37 \] \[ (q \circ r)(-4) = -19 \]
