Let's solve the problems step-by-step:
Given the functions:
[tex]\[ q(x) = -x + 2 \][/tex]
[tex]\[ r(x) = x^2 + 1 \][/tex]
1. Find [tex]\((q \circ r)(-1)\)[/tex]:
The notation [tex]\((q \circ r)(-1)\)[/tex] means that we first apply the function [tex]\( r \)[/tex] to [tex]\(-1\)[/tex] and then apply the function [tex]\( q \)[/tex] to the result.
### Step-by-Step Solution:
- Compute [tex]\( r(-1) \)[/tex]:
[tex]\[ r(-1) = (-1)^2 + 1 = 1 + 1 = 2 \][/tex]
- Now, apply [tex]\( q \)[/tex] to the result of [tex]\( r(-1) \)[/tex]:
[tex]\[ q(2) = -2 + 2 = 0 \][/tex]
So, [tex]\((q \circ r)(-1) = 0\)[/tex].
2. Find [tex]\((r \circ q)(-1)\)[/tex]:
The notation [tex]\((r \circ q)(-1)\)[/tex] means that we first apply the function [tex]\( q \)[/tex] to [tex]\(-1\)[/tex] and then apply the function [tex]\( r \)[/tex] to the result.
### Step-by-Step Solution:
- Compute [tex]\( q(-1) \)[/tex]:
[tex]\[ q(-1) = -(-1) + 2 = 1 + 2 = 3 \][/tex]
- Now, apply [tex]\( r \)[/tex] to the result of [tex]\( q(-1) \)[/tex]:
[tex]\[ r(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
So, [tex]\((r \circ q)(-1) = 10\)[/tex].
### Final Answer:
[tex]\[
\begin{array}{l}
(q \circ r)(-1) = 0 \\
(r \circ q)(-1) = 10 \\
\end{array}
\][/tex]
