Suppose that the functions [tex]$f$[/tex] and [tex]$g$[/tex] are defined for all real numbers [tex]$x$[/tex] as follows:

[tex]\[
\begin{array}{l}
f(x) = 4x \\
g(x) = 3x^2
\end{array}
\][/tex]

Write the expressions for [tex]$(g \cdot f)(x)$[/tex] and [tex]$(g - f)(x)$[/tex] and evaluate [tex]$(g + f)(-2)$[/tex].

[tex]\[
\begin{array}{l}
(g \cdot f)(x) = \square \\
(g - f)(x) = \square \\
(g + f)(-2) = \square
\end{array}
\][/tex]



Answer :

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]