To solve for [tex]\( g(f(x)) \)[/tex], we'll follow these steps:
1. Determine the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 2x
\][/tex]
2. Determine the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x^2 + 3
\][/tex]
3. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
We need to find [tex]\( g(f(x)) \)[/tex]. This means we will substitute the entire expression of [tex]\( f(x) \)[/tex] into the function [tex]\( g(x) \)[/tex].
So, let's substitute [tex]\( 2x \)[/tex] (which is [tex]\( f(x) \)[/tex]) into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(2x)
\][/tex]
4. Evaluate [tex]\( g(2x) \)[/tex] using the definition of [tex]\( g(x) \)[/tex]:
[tex]\[
g(2x) = (2x)^2 + 3
\][/tex]
5. Simplify the expression [tex]\( (2x)^2 + 3 \)[/tex]:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
Thus,
[tex]\[
(2x)^2 + 3 = 4x^2 + 3
\][/tex]
Therefore, [tex]\( g(f(x)) = 4x^2 + 3 \)[/tex].
The correct answer is
[tex]\[
4x^2 + 3
\][/tex]
