Let's work through the problem step by step.
Given the functions:
[tex]\[
\begin{array}{l}
g(x) = x + 6 \\
h(x) = 2x + 4
\end{array}
\][/tex]
1. Finding the expression for [tex]\((g \cdot h)(x)\)[/tex]:
The expression [tex]\((g \cdot h)(x)\)[/tex] represents the product of the functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex].
[tex]\[
(g \cdot h)(x) = g(x) \times h(x)
\][/tex]
Substituting the given functions:
[tex]\[
(g \cdot h)(x) = (x + 6) \times (2x + 4)
\][/tex]
2. Finding the expression for [tex]\((g + h)(x)\)[/tex]:
The expression [tex]\((g + h)(x)\)[/tex] represents the sum of the functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex].
[tex]\[
(g + h)(x) = g(x) + h(x)
\][/tex]
Substituting the given functions:
[tex]\[
(g + h)(x) = (x + 6) + (2x + 4) = 3x + 10
\][/tex]
3. Evaluating [tex]\((g - h)(2)\)[/tex]:
The expression [tex]\((g - h)(x)\)[/tex] represents the difference between the functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex].
[tex]\[
(g - h)(x) = g(x) - h(x)
\][/tex]
Specifically, we need to evaluate this difference at [tex]\(x = 2\)[/tex]:
[tex]\[
(g - h)(2) = g(2) - h(2)
\][/tex]
Substituting [tex]\(x = 2\)[/tex] into the given functions:
[tex]\[
g(2) = 2 + 6 = 8
\][/tex]
[tex]\[
h(2) = 2 \times 2 + 4 = 4 + 4 = 8
\][/tex]
Therefore,
[tex]\[
(g - h)(2) = 8 - 8 = 0
\][/tex]
To summarize:
[tex]\[
\begin{array}{l}
(g \cdot h)(x) = (x + 6) \times (2x + 4) \\
(g + h)(x) = 3x + 10 \\
(g - h)(2) = 0
\end{array}
\][/tex]
