To determine the midpoint of the [tex]\(x\)[/tex]-intercepts of the function [tex]\( f(x) = (x - 4)(x + 4) \)[/tex], we need to follow these steps:
1. Find the [tex]\(x\)[/tex]-intercepts:
The [tex]\(x\)[/tex]-intercepts of a function are the points where the graph of the function crosses the [tex]\(x\)[/tex]-axis. These points occur where [tex]\( f(x) = 0 \)[/tex].
[tex]\[
f(x) = (x - 4)(x + 4)
\][/tex]
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
(x - 4)(x + 4) = 0
\][/tex]
This equation tells us that the product of two factors is zero. For the product to be zero, at least one of the factors must be zero. Thus, we solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solving these equations gives us:
[tex]\[
x = 4 \quad \text{and} \quad x = -4
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex].
2. Calculate the midpoint:
The midpoint of two points on a number line is calculated by finding the average of the two coordinates. For the [tex]\( x \)[/tex]-intercepts [tex]\( x_1 = 4 \)[/tex] and [tex]\( x_2 = -4 \)[/tex], the midpoint [tex]\( M \)[/tex] is calculated as follows:
[tex]\[
M = \frac{x_1 + x_2}{2}
\][/tex]
Plugging in the values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
[tex]\[
M = \frac{4 + (-4)}{2} = \frac{0}{2} = 0
\][/tex]
Thus, the midpoint of the [tex]\(x\)[/tex]-intercepts of the function [tex]\( f(x) = (x - 4)(x + 4) \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the correct option is:
[tex]\[
(0, 0)
\][/tex]
