Let's start by defining the functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[
g(x) = 2x - 3
\][/tex]
[tex]\[
h(x) = 6x
\][/tex]
Now, we need to find the expressions for [tex]\( (h - g)(x) \)[/tex] and [tex]\( (h \cdot g)(x) \)[/tex], and evaluate [tex]\( (h + g)(1) \)[/tex].
1. Expression for [tex]\( (h - g)(x) \)[/tex]:
[tex]\[
(h - g)(x) = h(x) - g(x)
\][/tex]
Substitute the given functions [tex]\( h(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(h - g)(x) = 6x - (2x - 3)
\][/tex]
Distribute the negative sign inside the parentheses:
[tex]\[
(h - g)(x) = 6x - 2x + 3
\][/tex]
Simplify the expression:
[tex]\[
(h - g)(x) = 4x + 3
\][/tex]
So, we have:
[tex]\[
(h - g)(x) = 4x + 3
\][/tex]
2. Expression for [tex]\( (h \cdot g)(x) \)[/tex]:
[tex]\[
(h \cdot g)(x) = h(x) \cdot g(x)
\][/tex]
Substitute the given functions [tex]\( h(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
(h \cdot g)(x) = 6x \cdot (2x - 3)
\][/tex]
Distribute [tex]\( 6x \)[/tex]:
[tex]\[
(h \cdot g)(x) = 12x^2 - 18x
\][/tex]
So, we have:
[tex]\[
(h \cdot g)(x) = 12x^2 - 18x
\][/tex]
3. Evaluate [tex]\( (h + g)(1) \)[/tex]:
[tex]\[
(h + g)(1) = h(1) + g(1)
\][/tex]
First, evaluate [tex]\( h(1) \)[/tex]:
[tex]\[
h(1) = 6 \cdot 1 = 6
\][/tex]
Next, evaluate [tex]\( g(1) \)[/tex]:
[tex]\[
g(1) = 2 \cdot 1 - 3 = 2 - 3 = -1
\][/tex]
Now sum these values:
[tex]\[
(h + g)(1) = 6 + (-1) = 5
\][/tex]
So, we have:
[tex]\[
(h + g)(1) = 5
\][/tex]
To summarize, the final answers are:
[tex]\[
\begin{array}{l}
(h - g)(x) = 4x + 3 \\
(h \cdot g)(x) = 12x^2 - 18x \\
(h + g)(1) = 5
\end{array}
\][/tex]
