Let's address each part of the question step by step.
### Part 1: Expression for [tex]\((g \cdot f)(x)\)[/tex]
The composition of two functions [tex]\(g\)[/tex] and [tex]\(f\)[/tex], denoted by [tex]\((g \cdot f)(x)\)[/tex], is calculated by multiplying the output of [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex].
Given:
[tex]\[ f(x) = 4x \][/tex]
[tex]\[ g(x) = 3x^2 \][/tex]
Therefore,
[tex]\[ (g \cdot f)(x) = g(x) \cdot f(x) = (3x^2) \cdot (4x) \][/tex]
Multiplying these together:
[tex]\[ (g \cdot f)(x) = 12x^3 \][/tex]
### Part 2: Expression for [tex]\((g-f)(x)\)[/tex]
The difference of two functions [tex]\(g\)[/tex] and [tex]\(f\)[/tex], denoted by [tex]\((g-f)(x)\)[/tex], is calculated by subtracting [tex]\(f(x)\)[/tex] from [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = 4x \][/tex]
[tex]\[ g(x) = 3x^2 \][/tex]
Therefore,
[tex]\[ (g - f)(x) = g(x) - f(x) = 3x^2 - 4x \][/tex]
### Part 3: Evaluate [tex]\((g+f)(-2)\)[/tex]
The sum of two functions [tex]\(g\)[/tex] and [tex]\(f\)[/tex], denoted by [tex]\((g+f)(x)\)[/tex], is calculated by adding [tex]\(g(x)\)[/tex] and [tex]\(f(x)\)[/tex]. We then need to evaluate this sum at [tex]\(x = -2\)[/tex].
Given:
[tex]\[ f(x) = 4x \][/tex]
[tex]\[ g(x) = 3x^2 \][/tex]
Evaluate:
[tex]\[ f(-2) = 4 \cdot (-2) = -8 \][/tex]
[tex]\[ g(-2) = 3 \cdot (-2)^2 = 3 \cdot 4 = 12 \][/tex]
Therefore,
[tex]\[ (g + f)(-2) = g(-2) + f(-2) = 12 + (-8) = 4 \][/tex]
### Summary of the Results:
[tex]\[
\begin{array}{c}
(g \cdot f)(x) = 12x^3 \\
(g - f)(x) = 3x^2 - 4x \\
(g + f)(-2) = 4 \\
\end{array}
\][/tex]
