To solve the problems given, we need to perform the following:
1. Find the function [tex]\((f - g)(x)\)[/tex].
2. Find the function [tex]\((f + g)(x)\)[/tex].
3. Evaluate the result of [tex]\((f \cdot g)(-2)\)[/tex].
Here are the steps to solve each part:
### Step 1: Find [tex]\((f - g)(x)\)[/tex]
The function [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are given as:
[tex]\[ f(x) = x - 1 \][/tex]
[tex]\[ g(x) = 3x^2 \][/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 the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = (x - 1) - 3x^2 \][/tex]
Therefore:
[tex]\[ (f - g)(x) = -3x^2 + x - 1 \][/tex]
### Step 2: Find [tex]\((f + g)(x)\)[/tex]
To find [tex]\((f + g)(x)\)[/tex], we add [tex]\(g(x)\)[/tex] to [tex]\(f(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f + g)(x) = (x - 1) + 3x^2 \][/tex]
Therefore:
[tex]\[ (f + g)(x) = 3x^2 + x - 1 \][/tex]
### Step 3: Evaluate [tex]\((f \cdot g)(-2)\)[/tex]
To find [tex]\((f \cdot g)(-2)\)[/tex], we first find [tex]\(f(-2)\)[/tex] and [tex]\(g(-2)\)[/tex], then multiply the results.
Find [tex]\(f(-2)\)[/tex]:
[tex]\[ f(-2) = (-2) - 1 = -3 \][/tex]
Find [tex]\(g(-2)\)[/tex]:
[tex]\[ g(-2) = 3(-2)^2 = 3 \cdot 4 = 12 \][/tex]
Now multiply the results:
[tex]\[ (f \cdot g)(-2) = f(-2) \cdot g(-2) \][/tex]
[tex]\[ (f \cdot g)(-2) = (-3) \cdot 12 = -36 \][/tex]
#### Final Results
[tex]\[
\begin{array}{l}
(f - g)(x) = -3x^2 + x - 1 \\
(f + g)(x) = 3x^2 + x - 1 \\
(f \cdot g)(-2) = -36 \\
\end{array}
\][/tex]
