To determine the value of [tex]\((f - g)(2)\)[/tex], we first need to find the individual values of [tex]\(f(2)\)[/tex] and [tex]\(g(2)\)[/tex].
### Step 1: Calculate [tex]\(f(2)\)[/tex]
The function [tex]\(f(x)\)[/tex] is defined as:
[tex]\[ f(x) = 3x^2 + 1 \][/tex]
Plugging in [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 3(2)^2 + 1 \][/tex]
[tex]\[ f(2) = 3 \times 4 + 1 \][/tex]
[tex]\[ f(2) = 12 + 1 \][/tex]
[tex]\[ f(2) = 13 \][/tex]
### Step 2: Calculate [tex]\(g(2)\)[/tex]
The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[ g(x) = 1 - x \][/tex]
Plugging in [tex]\(x = 2\)[/tex]:
[tex]\[ g(2) = 1 - 2 \][/tex]
[tex]\[ g(2) = -1 \][/tex]
### Step 3: Calculate [tex]\((f - g)(2)\)[/tex]
The expression [tex]\((f - g)(x)\)[/tex] represents the difference between the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x\)[/tex], which is given by:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
So, we need to find [tex]\((f - g)(2)\)[/tex]:
[tex]\[ (f - g)(2) = f(2) - g(2) \][/tex]
Substituting the values we previously found:
[tex]\[ (f - g)(2) = 13 - (-1) \][/tex]
[tex]\[ (f - g)(2) = 13 + 1 \][/tex]
[tex]\[ (f - g)(2) = 14 \][/tex]
Thus, the value of [tex]\((f - g)(2)\)[/tex] is [tex]\(\boxed{14}\)[/tex].
