To find the midpoint of the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = (x-2)(x-4) \)[/tex], follow these steps:
1. Identify the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts of a function occur where the function equals zero. To find them, we set [tex]\( f(x) = 0 \)[/tex] and solve the equation:
[tex]\[
(x - 2)(x - 4) = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
This equation is satisfied when either [tex]\( x - 2 = 0 \)[/tex] or [tex]\( x - 4 = 0 \)[/tex]. Solving these:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts are at [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex].
3. Find the midpoint of the [tex]\( x \)[/tex]-intercepts:
To find the midpoint of these points, we use the midpoint formula for two points, [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex], which is given by:
[tex]\[
\text{Midpoint} = \frac{x_1 + x_2}{2}
\][/tex]
Here, [tex]\( x_1 = 2 \)[/tex] and [tex]\( x_2 = 4 \)[/tex]. Substituting these values into the formula:
[tex]\[
\text{Midpoint} = \frac{2 + 4}{2} = \frac{6}{2} = 3
\][/tex]
4. Result:
Therefore, the midpoint of the [tex]\( x \)[/tex]-intercepts is [tex]\( 3 \)[/tex].
Given the options provided, the correct answer is:
[tex]\[
(3,0)
\][/tex]
