To find the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = \frac{(x-20)(x+9)}{x(x-12)} \)[/tex], we need to determine where the function crosses the [tex]\( x \)[/tex]-axis. This occurs when the numerator is zero and the denominator is non-zero.
The numerator of the function is [tex]\((x-20)(x+9)\)[/tex]. We set this equal to zero to find the [tex]\( x \)[/tex]-intercepts.
[tex]\[
(x-20)(x+9) = 0
\][/tex]
This equation will be zero when either [tex]\( x-20 = 0 \)[/tex] or [tex]\( x+9 = 0 \)[/tex].
1. Solving [tex]\( x-20 = 0 \)[/tex]:
[tex]\[
x = 20
\][/tex]
2. Solving [tex]\( x+9 = 0 \)[/tex]:
[tex]\[
x = -9
\][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts of the function are [tex]\( x = 20 \)[/tex] and [tex]\( x = -9 \)[/tex].
Finally, we identify the smaller and larger intercepts:
- Smaller intercept at [tex]\( x = -9 \)[/tex]
- Larger intercept at [tex]\( x = 20 \)[/tex]
Therefore:
Smaller intercept at [tex]\( x = -9 \)[/tex] [tex]\(\square\)[/tex]
Larger intercept at [tex]\( x = 20 \)[/tex] [tex]\(\square\)[/tex]
