Let's find [tex]\((q \circ r)(-2)\)[/tex] and [tex]\((r \circ q)(-2)\)[/tex] using the definitions of the functions [tex]\(q(x) = 5x\)[/tex] and [tex]\(r(x) = 4x + 4\)[/tex].
### Finding [tex]\((q \circ r)(-2)\)[/tex]
The composition [tex]\( (q \circ r)(x) \)[/tex] means we first apply the function [tex]\( r \)[/tex] to [tex]\( x \)[/tex], and then apply the function [tex]\( q \)[/tex] to the result of [tex]\( r(x) \)[/tex].
1. Calculate [tex]\( r(-2) \)[/tex]:
[tex]\[
r(-2) = 4(-2) + 4 = -8 + 4 = -4
\][/tex]
2. Apply [tex]\( q \)[/tex] to the result of [tex]\( r(-2) \)[/tex]:
[tex]\[
q(r(-2)) = q(-4) = 5(-4) = -20
\][/tex]
So, [tex]\((q \circ r)(-2) = -20\)[/tex].
### Finding [tex]\((r \circ q)(-2)\)[/tex]
The composition [tex]\( (r \circ q)(x) \)[/tex] means we first apply the function [tex]\( q \)[/tex] to [tex]\( x \)[/tex], and then apply the function [tex]\( r \)[/tex] to the result of [tex]\( q(x) \)[/tex].
1. Calculate [tex]\( q(-2) \)[/tex]:
[tex]\[
q(-2) = 5(-2) = -10
\][/tex]
2. Apply [tex]\( r \)[/tex] to the result of [tex]\( q(-2) \)[/tex]:
[tex]\[
r(q(-2)) = r(-10) = 4(-10) + 4 = -40 + 4 = -36
\][/tex]
So, [tex]\((r \circ q)(-2) = -36\)[/tex].
### Conclusion
[tex]\[
(q \circ r)(-2) = -20
\][/tex]
[tex]\[
(r \circ q)(-2) = -36
\][/tex]
