Graph the rational function.
-$\begin{aligned}
\mathrm { y } = \frac { \mathrm { x } } { \mathrm { x } ^ { 2 } - 1 } \
\end{aligned}$

Question

Quizplus · Accepted Answer

The denominator of the function is $x^2-1$, which can be factored into $(x-1)(x+1)$. This means that the function has vertical asymptotes at $x=1$ and $x=-1$. To find the x-intercept, set $y=0$:
$$\frac{x}{x^2-1}=0$$
$$x=0$$
Therefore, the function has an x-intercept at $(0,0)$. To determine the sign of the function on either side of the vertical asymptotes, choose some test points in each region and evaluate the function at those points. For example, for values of x greater than 1, choose x=2:
$$\frac{2}{2^2-1}=\frac{2}{3}$$
Since the numerator and denominator have the same sign, the function is positive. For values of x between 1 and -1, choose x=0:
$$\frac{0}{0^2-1}=0$$
The function is zero. For values of x less than -1, choose x=-2:
$$\frac{-2}{(-2)^2-1}=-\frac{2}{3}$$
The numerator and denominator have opposite signs, so the function is negative. Putting all of this information together, we can sketch the graph of the function as a hyperbola with vertical asymptotes at x=-1 and x=1, an x-intercept at (0,0), and passing through the point (-1,-1).

Graph the Rational Function y=xx2−1\begin{aligned}\mathrm { y } = \frac { \mathrm { x } } { \mathrm { x } ^ { 2 } - 1 } \\\end{aligned}y=x2−1x​​

Graph the Rational Function $\begin{aligned}\mathrm { y } = \frac { \mathrm { x } } { \mathrm { x } ^ { 2 } - 1 } \\\end{aligned}$