To determine the [tex]\( x \)[/tex]-intercept of the rational function [tex]\( f(x) = \frac{x}{6x - 1} \)[/tex], we need to find the value of [tex]\( x \)[/tex] where the function equals zero. This occurs when the numerator of the rational function is zero, provided that the denominator is not zero at the same point.
Let's go through the steps to find the [tex]\( x \)[/tex]-intercept:
1. Set the function equal to zero:
[tex]\[
f(x) = \frac{x}{6x - 1} = 0
\][/tex]
2. Solve the equation [tex]\( \frac{x}{6x - 1} = 0 \)[/tex]:
For a rational function [tex]\(\frac{a}{b}\)[/tex] to be zero, the numerator [tex]\(a\)[/tex] must be zero while the denominator [tex]\(b\)[/tex] must be non-zero.
3. Set the numerator equal to zero:
[tex]\[
x = 0
\][/tex]
4. Check the denominator to ensure it is not zero when [tex]\( x = 0 \)[/tex]:
[tex]\[
6x - 1 \neq 0 \quad \text{when} \quad x = 0
\][/tex]
Substituting [tex]\( x = 0 \)[/tex] into the denominator:
[tex]\[
6(0) - 1 = -1 \neq 0
\][/tex]
Since the denominator is [tex]\(-1\)[/tex] (which is not zero), it confirms that [tex]\( x = 0 \)[/tex] is a valid solution.
5. State the [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept is the point where the function crosses the [tex]\( x \)[/tex]-axis. This happens at [tex]\( x = 0 \)[/tex].
Therefore, the [tex]\( x \)[/tex]-intercept of the function [tex]\( f(x) = \frac{x}{6x - 1} \)[/tex] is:
[tex]\[
(0, 0)
\][/tex]
So, the correct answer is:
[tex]\[
(0, 0)
\][/tex]
