Sure, let's evaluate the compositions of the functions [tex]\( q(x) \)[/tex] and [tex]\( r(x) \)[/tex] at [tex]\( x = -1 \)[/tex] step-by-step.
First, we need to recall the definitions of the functions:
[tex]\[
q(x) = 2x - 2
\][/tex]
[tex]\[
r(x) = -x^2 + 1
\][/tex]
### Step 1: Finding [tex]\((r \circ q)(-1)\)[/tex]
The notation [tex]\((r \circ q)(-1)\)[/tex] represents the composition of [tex]\( r \)[/tex] and [tex]\( q \)[/tex], evaluated at [tex]\( x = -1 \)[/tex]. This means we first apply [tex]\( q \)[/tex] to [tex]\( -1 \)[/tex] and then apply [tex]\( r \)[/tex] to the result of [tex]\( q(-1) \)[/tex].
1. Calculate [tex]\( q(-1) \)[/tex]:
[tex]\[
q(-1) = 2(-1) - 2 = -2 - 2 = -4
\][/tex]
2. Substitute [tex]\( q(-1) \)[/tex] into [tex]\( r \)[/tex]:
[tex]\[
r(q(-1)) = r(-4)
\][/tex]
3. Calculate [tex]\( r(-4) \)[/tex]:
[tex]\[
r(-4) = -(-4)^2 + 1 = -(16) + 1 = -15
\][/tex]
Therefore,
[tex]\[
(r \circ q)(-1) = -15
\][/tex]
### Step 2: Finding [tex]\((q \circ r)(-1)\)[/tex]
The notation [tex]\((q \circ r)(-1)\)[/tex] represents the composition of [tex]\( q \)[/tex] and [tex]\( r \)[/tex], evaluated at [tex]\( x = -1 \)[/tex]. This means we first apply [tex]\( r \)[/tex] to [tex]\( -1 \)[/tex] and then apply [tex]\( q \)[/tex] to the result of [tex]\( r(-1) \)[/tex].
1. Calculate [tex]\( r(-1) \)[/tex]:
[tex]\[
r(-1) = -(-1)^2 + 1 = -1 + 1 = 0
\][/tex]
2. Substitute [tex]\( r(-1) \)[/tex] into [tex]\( q \)[/tex]:
[tex]\[
q(r(-1)) = q(0)
\][/tex]
3. Calculate [tex]\( q(0) \)[/tex]:
[tex]\[
q(0) = 2(0) - 2 = 0 - 2 = -2
\][/tex]
Therefore,
[tex]\[
(q \circ r)(-1) = -2
\][/tex]
### Final Answers:
[tex]\[
(r \circ q)(-1) = -15
\][/tex]
[tex]\[
(q \circ r)(-1) = -2
\][/tex]
