Certainly! Let's solve the problem step-by-step.
1. We have the function [tex]\( f(g) = 2g + 2 \)[/tex].
2. We also have the function [tex]\( g(x) = -2x \)[/tex].
3. We need to find the composition of these functions, specifically [tex]\( f(g(x)) \)[/tex].
4. To do this, we first determine [tex]\( g(x) \)[/tex]. From the second function, we know:
[tex]\[
g(x) = -2x
\][/tex]
5. Next, we substitute [tex]\( g(x) \)[/tex] into the function [tex]\( f(g) \)[/tex]. Given [tex]\( g(x) = -2x \)[/tex], we replace [tex]\( g \)[/tex] in [tex]\( f(g) = 2g + 2 \)[/tex] with [tex]\(-2x\)[/tex]:
[tex]\[
f(g(x)) = f(-2x)
\][/tex]
6. Now, we return to the original function [tex]\( f(g) \)[/tex], which is [tex]\( 2g + 2 \)[/tex]. We substitute [tex]\( -2x \)[/tex] for [tex]\( g \)[/tex]:
[tex]\[
f(-2x) = 2(-2x) + 2
\][/tex]
7. Simplify the expression inside the function:
[tex]\[
2(-2x) + 2 = -4x + 2
\][/tex]
Therefore, the composed function [tex]\( f(g(x)) \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[
f(g(x)) = -4x + 2
\][/tex]
This is the final expression for [tex]\( f(g(x)) \)[/tex] based on the given functions.
