To determine 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 occur where the function [tex]\( f(x) \)[/tex] equals zero. Thus, we need to solve [tex]\( (x - 2)(x - 4) = 0 \)[/tex].
Setting each factor equal to zero gives us the [tex]\( x \)[/tex]-intercepts:
[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 [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Calculate the midpoint:
The midpoint of two points [tex]\( a \)[/tex] and [tex]\( b \)[/tex] on the number line is given by [tex]\( \frac{a + b}{2} \)[/tex].
Substituting [tex]\( a = 2 \)[/tex] and [tex]\( b = 4 \)[/tex]:
[tex]\[
\text{Midpoint} = \frac{2 + 4}{2} = \frac{6}{2} = 3
\][/tex]
Therefore, the coordinates of the midpoint are [tex]\( (3,0) \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{(3,0)}
\][/tex]
