Sure, let’s work through this step-by-step.
### Given Functions:
[tex]\[ q(x) = -5x - 1 \][/tex]
[tex]\[ r(x) = -3x \][/tex]
### Find [tex]\( (r \circ q)(-2) \)[/tex]:
The composition of [tex]\( r \)[/tex] and [tex]\( q \)[/tex] evaluated at [tex]\(-2\)[/tex] means we first apply [tex]\( q \)[/tex] to [tex]\(-2\)[/tex] and then apply [tex]\( r \)[/tex] to the result.
1. Calculate [tex]\( q(-2) \)[/tex]:
[tex]\[
q(-2) = -5(-2) - 1 = 10 - 1 = 9
\][/tex]
2. With [tex]\( q(-2) = 9 \)[/tex], now calculate [tex]\( r(q(-2)) \)[/tex], which is the same as [tex]\( r(9) \)[/tex]:
[tex]\[
r(9) = -3(9) = -27
\][/tex]
Therefore, [tex]\( (r \circ q)(-2) = -27 \)[/tex].
### Find [tex]\( (q \circ r)(-2) \)[/tex]:
The composition of [tex]\( q \)[/tex] and [tex]\( r \)[/tex] evaluated at [tex]\(-2\)[/tex] means we first apply [tex]\( r \)[/tex] to [tex]\(-2\)[/tex] and then apply [tex]\( q \)[/tex] to the result.
1. Calculate [tex]\( r(-2) \)[/tex]:
[tex]\[
r(-2) = -3(-2) = 6
\][/tex]
2. With [tex]\( r(-2) = 6 \)[/tex], now calculate [tex]\( q(r(-2)) \)[/tex], which is the same as [tex]\( q(6) \)[/tex]:
[tex]\[
q(6) = -5(6) - 1 = -30 - 1 = -31
\][/tex]
Therefore, [tex]\( (q \circ r)(-2) = -31 \)[/tex].
### Summary:
[tex]\[
(r \circ q)(-2) = -27
\][/tex]
[tex]\[
(q \circ r)(-2) = -31
\][/tex]
