Let's start by defining the given functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[ g(x) = 3x - 4 \][/tex]
[tex]\[ h(x) = x - 3 \][/tex]
### 1. Expression for [tex]\((h + g)(x)\)[/tex]
To find [tex]\((h + g)(x)\)[/tex], we simply add the two functions [tex]\( h(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (h + g)(x) = h(x) + g(x) \][/tex]
[tex]\[ (h + g)(x) = (x - 3) + (3x - 4) \][/tex]
Combine like terms:
[tex]\[ (h + g)(x) = x + 3x - 3 - 4 \][/tex]
[tex]\[ (h + g)(x) = 4x - 7 \][/tex]
### 2. Expression for [tex]\((h \cdot g)(x)\)[/tex]
To find [tex]\((h \cdot g)(x)\)[/tex], we multiply the two functions [tex]\( h(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (h \cdot g)(x) = h(x) \cdot g(x) \][/tex]
[tex]\[ (h \cdot g)(x) = (x - 3) \cdot (3x - 4) \][/tex]
### 3. Evaluation of [tex]\((h - g)(2)\)[/tex]
To find [tex]\((h - g)(2)\)[/tex], we need to evaluate the two functions at [tex]\( x = 2 \)[/tex] and then subtract the results:
[tex]\[ (h - g)(2) = h(2) - g(2) \][/tex]
Evaluate [tex]\( h(2) \)[/tex]:
[tex]\[ h(2) = 2 - 3 \][/tex]
[tex]\[ h(2) = -1 \][/tex]
Evaluate [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = 3 \cdot 2 - 4 \][/tex]
[tex]\[ g(2) = 6 - 4 \][/tex]
[tex]\[ g(2) = 2 \][/tex]
Subtract the results:
[tex]\[ (h - g)(2) = -1 - 2 \][/tex]
[tex]\[ (h - g)(2) = -3 \][/tex]
So, the final results are:
[tex]\[
\begin{array}{c}
(h + g)(x) = 4x - 7 \\
(h \cdot g)(x) = (x - 3)(3x - 4) \\
(h - g)(2) = -3
\end{array}
\][/tex]
