To evaluate [tex]\((mn)(x)\)[/tex] for [tex]\( x = -3 \)[/tex], we need to follow these steps:
1. Evaluate [tex]\( m(x) \)[/tex]:
Given [tex]\( m(x) = x + 2 \)[/tex],
[tex]\[
m(-3) = -3 + 2 = -1.
\][/tex]
2. Evaluate [tex]\( n(x) \)[/tex]:
Given [tex]\( n(x) = 2x \)[/tex],
[tex]\[
n(-3) = 2 \cdot (-3) = -6.
\][/tex]
3. Evaluate [tex]\((mn)(x)\)[/tex]:
By definition, [tex]\((mn)(x)\)[/tex] means [tex]\( m(n(x)) \)[/tex].
Substituting the value of [tex]\( n(-3) \)[/tex] into [tex]\( m \)[/tex]:
[tex]\[
m(n(-3)) = m(-6).
\][/tex]
4. Evaluate [tex]\( m(-6) \)[/tex]:
Using the function [tex]\( m(x) \)[/tex],
[tex]\[
m(-6) = -6 + 2 = -4.
\][/tex]
So, the value of [tex]\((mn)(-3)\)[/tex] is
[tex]\[
\boxed{-4}.
\][/tex]
Thus, the evaluated results for the given [tex]\( x = -3 \)[/tex] are:
- [tex]\( m(-3) = -1 \)[/tex]
- [tex]\( n(-3) = -6 \)[/tex]
- [tex]\( (mn)(-3) = -4 \)[/tex].
The final result for [tex]\( (mn)(-3) \)[/tex] is
[tex]\[
\boxed{-4}.
\][/tex]
