To find [tex]\((g \circ f)(x)\)[/tex], we need to apply the function [tex]\(f(x)\)[/tex] first and then take the result and apply the function [tex]\(g\)[/tex] to that. Let's proceed step-by-step.
1. Define the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = 2x + 1
\][/tex]
2. Apply the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = 2x + 1
\][/tex]
3. Now, take the result of [tex]\(f(x)\)[/tex] (which is [tex]\(2x + 1\)[/tex]) and apply the function [tex]\(g\)[/tex] to it:
[tex]\[
g(x) = 6x + 2
\][/tex]
4. Replace [tex]\(x\)[/tex] in [tex]\(g(x)\)[/tex] with [tex]\(f(x)\)[/tex]:
[tex]\[
g(f(x)) = g(2x + 1)
\][/tex]
5. Apply [tex]\(g\)[/tex] to [tex]\(2x + 1\)[/tex]:
[tex]\[
g(2x + 1) = 6(2x + 1) + 2
\][/tex]
6. Distribute and simplify:
[tex]\[
g(2x + 1) = 6 \cdot 2x + 6 \cdot 1 + 2
\][/tex]
[tex]\[
= 12x + 6 + 2
\][/tex]
[tex]\[
= 12x + 8
\][/tex]
Thus, the composition [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[
(g \circ f)(x) = 12x + 8
\][/tex]
