Given the functions [tex]\(g(x) = 3x - 4\)[/tex] and [tex]\(h(x) = x - 3\)[/tex], we need to find the expressions for [tex]\((h + g)(x)\)[/tex] and [tex]\((h \cdot g)(x)\)[/tex], and evaluate [tex]\((h - g)(2)\)[/tex].
1. Finding [tex]\((h + g)(x)\)[/tex]:
[tex]\[
(h + g)(x) = h(x) + g(x)
\][/tex]
We are given [tex]\(h(x) = x - 3\)[/tex] and [tex]\(g(x) = 3x - 4\)[/tex]. Therefore,
[tex]\[
h(x) + g(x) = (x - 3) + (3x - 4)
\][/tex]
Simplify the expression by combining like terms:
[tex]\[
(h + g)(x) = x - 3 + 3x - 4 = 4x - 7
\][/tex]
2. Finding [tex]\((h \cdot g)(x)\)[/tex]:
[tex]\[
(h \cdot g)(x) = h(x) \cdot g(x)
\][/tex]
Using the given functions [tex]\(h(x) = x - 3\)[/tex] and [tex]\(g(x) = 3x - 4\)[/tex],
[tex]\[
h(x) \cdot g(x) = (x - 3) \cdot (3x - 4)
\][/tex]
This is the product of two binomials. The expression can be left in its factored form as desired:
[tex]\[
(h \cdot g)(x) = (x - 3) (3x - 4)
\][/tex]
3. Evaluating [tex]\((h - g)(2)\)[/tex]:
[tex]\[
(h - g)(x) = h(x) - g(x)
\][/tex]
Substitute [tex]\(h(x) = x - 3\)[/tex] and [tex]\(g(x) = 3x - 4\)[/tex]:
[tex]\[
h(x) - g(x) = (x - 3) - (3x - 4)
\][/tex]
Simplify this expression:
[tex]\[
h(x) - g(x) = x - 3 - 3x + 4 = -2x + 1
\][/tex]
Now, evaluate this at [tex]\(x = 2\)[/tex]:
[tex]\[
(h - g)(2) = -2(2) + 1 = -4 + 1 = -3
\][/tex]
Final answers:
[tex]\[
\begin{array}{r}
(h + g)(x) = 4x - 7 \\
(h \cdot g)(x) = (x - 3) (3x - 4) \\
(h - g)(2) = -3
\end{array}
\][/tex]
