To solve for [tex]\((f - g)(2)\)[/tex], we need to understand that this expression represents the difference between the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] evaluated at [tex]\(x = 2\)[/tex].
1. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
Recall the function [tex]\(f(x) = 3x^2 + 1\)[/tex].
Plug in [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = 3(2)^2 + 1
\][/tex]
Calculate the square of 2:
[tex]\[
(2)^2 = 4
\][/tex]
Multiply by 3:
[tex]\[
3 \times 4 = 12
\][/tex]
Add 1:
[tex]\[
12 + 1 = 13
\][/tex]
Therefore,
[tex]\[
f(2) = 13
\][/tex]
2. Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 2\)[/tex]:
Recall the function [tex]\(g(x) = 1 - x\)[/tex].
Plug in [tex]\(x = 2\)[/tex]:
[tex]\[
g(2) = 1 - 2
\][/tex]
Subtract 2 from 1:
[tex]\[
1 - 2 = -1
\][/tex]
Therefore,
[tex]\[
g(2) = -1
\][/tex]
3. Combine the results to find [tex]\((f - g)(2)\)[/tex]:
The expression [tex]\((f - g)(2)\)[/tex] is found by subtracting [tex]\(g(2)\)[/tex] from [tex]\(f(2)\)[/tex]:
[tex]\[
(f - g)(2) = f(2) - g(2)
\][/tex]
Substitute the values we calculated:
[tex]\[
(f - g)(2) = 13 - (-1)
\][/tex]
This simplifies to:
[tex]\[
13 + 1 = 14
\][/tex]
So, the value of [tex]\((f - g)(2)\)[/tex] is [tex]\(\boxed{14}\)[/tex].
