To solve for the value of [tex]\( (f - g)(2) \)[/tex] given the functions [tex]\( f(x) = 3x^2 + 1 \)[/tex] and [tex]\( g(x) = 1 - x \)[/tex], we will follow these steps:
1. Evaluate [tex]\( f(2) \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(2) = 3(2)^2 + 1
\][/tex]
- Calculate the square of 2:
[tex]\[
2^2 = 4
\][/tex]
- Multiply 4 by 3:
[tex]\[
3 \times 4 = 12
\][/tex]
- Add 1 to 12:
[tex]\[
12 + 1 = 13
\][/tex]
So, [tex]\( f(2) = 13 \)[/tex].
2. Evaluate [tex]\( g(2) \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(2) = 1 - 2
\][/tex]
- Calculate the subtraction:
[tex]\[
1 - 2 = -1
\][/tex]
So, [tex]\( g(2) = -1 \)[/tex].
3. Calculate [tex]\( (f - g)(2) \)[/tex]:
- To find [tex]\( (f - g)(x) \)[/tex], we subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
- Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
(f - g)(2) = f(2) - g(2)
\][/tex]
- Use the values we found:
[tex]\[
(f - g)(2) = 13 - (-1)
\][/tex]
- Simplify the expression:
[tex]\[
13 + 1 = 14
\][/tex]
Therefore, the value of [tex]\( (f - g)(2) \)[/tex] is [tex]\( 14 \)[/tex].
So, the correct answer is:
[tex]\[
14
\][/tex]
