Certainly! Let's solve this problem step-by-step.
1. Finding [tex]\((f \cdot g)(x)\)[/tex]:
The expression [tex]\( (f \cdot g)(x) \)[/tex] represents the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
[tex]\[
f(x) = 2x - 4
\][/tex]
[tex]\[
g(x) = x - 3
\][/tex]
Therefore,
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x - 4)(x - 3)
\][/tex]
To simplify this, we expand the product:
[tex]\[
(2x - 4)(x - 3) = 2x \cdot x + 2x \cdot (-3) - 4 \cdot x - 4 \cdot (-3)
\][/tex]
[tex]\[
= 2x^2 - 6x - 4x + 12
\][/tex]
Combine like terms:
[tex]\[
= 2x^2 - 10x + 12
\][/tex]
So,
[tex]\[
(f \cdot g)(x) = 2x^2 - 10x + 12
\][/tex]
2. Finding [tex]\((f+g)(x)\)[/tex]:
The expression [tex]\( (f + g)(x) \)[/tex] represents the sum of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
[tex]\[
f(x) = 2x - 4
\][/tex]
[tex]\[
g(x) = x - 3
\][/tex]
Therefore,
[tex]\[
(f + g)(x) = f(x) + g(x) = (2x - 4) + (x - 3)
\][/tex]
To simplify this, combine like terms:
[tex]\[
= 2x + x - 4 - 3
\][/tex]
[tex]\[
= 3x - 7
\][/tex]
So,
[tex]\[
(f + g)(x) = 3x - 7
\][/tex]
3. Evaluating [tex]\((f - g)(1)\)[/tex]:
The expression [tex]\( (f - g)(x) \)[/tex] represents the difference of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
So, we need to find [tex]\( (f - g)(1) \)[/tex]. First, let's find [tex]\( f(1) \)[/tex] and [tex]\( g(1) \)[/tex].
[tex]\[
f(x) = 2x - 4 \quad \text{so} \quad f(1) = 2(1) - 4 = 2 - 4 = -2
\][/tex]
[tex]\[
g(x) = x - 3 \quad \text{so} \quad g(1) = 1 - 3 = -2
\][/tex]
Therefore,
[tex]\[
(f - g)(1) = f(1) - g(1) = -2 - (-2)
\][/tex]
Simplify the expression by subtracting a negative value:
[tex]\[
= -2 + 2 = 0
\][/tex]
So, the final results are:
[tex]\[
\begin{array}{c}
(f \cdot g)(x)=2x^2 - 10x + 12 \\
(f+g)(x)=3x - 7 \\
(f-g)(1)=0
\end{array}
\][/tex]
