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 of the function:
The [tex]\( x \)[/tex]-intercepts occur where the function [tex]\( f(x) \)[/tex] equals zero. Set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
(x-2)(x-4) = 0
\][/tex]
This equation is satisfied when either [tex]\( x-2 = 0 \)[/tex] or [tex]\( x-4 = 0 \)[/tex].
Solving these equations:
[tex]\[
x - 2 = 0 \implies x = 2
\][/tex]
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Find the midpoint of these intercepts:
The midpoint of two points [tex]\( (x_1, 0) \)[/tex] and [tex]\( (x_2, 0) \)[/tex] can be found using the midpoint formula for the [tex]\( x \)[/tex]-coordinates:
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, 0 \right)
\][/tex]
Substitute the values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
[tex]\[
x_1 = 2, \quad x_2 = 4
\][/tex]
[tex]\[
\text{Midpoint} = \left( \frac{2 + 4}{2}, 0 \right) = \left( \frac{6}{2}, 0 \right) = (3, 0)
\][/tex]
Therefore, the midpoint of the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = (x-2)(x-4) \)[/tex] is [tex]\( \boxed{(3, 0)} \)[/tex].
