Question

multiple choice 8 points
which graph description best matches:
$f(x) = \frac{x-1}{x^2}$
○ a vertical asymptote at $x = 0$ and an x-intercept at $x = 1$
○ two vertical asymptotes at $x = \pm1$
○ a hole at $x = 0$
○ one vertical asymptote at $x = 1$

Explanation:

Step1: Find vertical asymptotes

Vertical asymptotes occur where the denominator is 0 and the numerator is not 0. Set $x^3=0$, so $x=0$. At $x=0$, numerator $0-1=-1
eq0$, so vertical asymptote at $x=0$.

Step2: Find x-intercept

x-intercept occurs where $f(x)=0$, so set numerator $x-1=0$. Solve for $x$: $x=1$.

Step3: Check for holes

Holes occur when a factor cancels in numerator/denominator. The numerator $x-1$ and denominator $x^3$ have no common factors, so no holes.

Answer:

A vertical asymptote at $x = 0$ and an x-intercept at $x = 1$