To solve for [tex]\((g \circ f)(3)\)[/tex], we need to follow these steps:
1. Evaluate [tex]\(f(3)\)[/tex].
2. Use the result of [tex]\(f(3)\)[/tex] to find [tex]\(g(f(3))\)[/tex].
Given the equations:
[tex]\[ f(x) = -2x + 8 \][/tex]
[tex]\[ g(x) = 3x \][/tex]
Step 1: Evaluate [tex]\(f(3)\)[/tex]
Substitute [tex]\(x = 3\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(3) = -2(3) + 8
\][/tex]
Perform the multiplication and addition:
[tex]\[
f(3) = -6 + 8 = 2
\][/tex]
So, [tex]\(f(3) = 2\)[/tex].
Step 2: Use the result of [tex]\(f(3)\)[/tex] to find [tex]\(g(f(3))\)[/tex]
Now, we need to evaluate [tex]\(g(2)\)[/tex], since [tex]\(f(3) = 2\)[/tex]:
[tex]\[
g(2) = 3(2)
\][/tex]
Perform the multiplication:
[tex]\[
g(2) = 6
\][/tex]
Therefore, [tex]\((g \circ f)(3) = g(f(3)) = g(2) = 6\)[/tex].
The correct answer is:
A. 6
