Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ f(x)=\frac{x}{x^{2}+x-2} $$

Short answer

The vertical asymptotes of the function are \(x=1\) and \(x=-2\).

Step-by-step solution

Identify the Function

Here, the function provided is \( f(x) = \frac{x} {x^{2}+x-2} \).

Find the Denominator and Set it to Zero

To find the vertical asymptotes, the denominator of this function must be equal to zero. So, we set \(x^{2}+x-2=0\).

Factor the Equation

Factoring the equation will give us the roots of the denominator. Factoring the equation \(x^{2}+x-2=0\) will give \((x-1)(x+2)=0\).

Find the Roots of the Equation

Setting each factor equal to zero will give the zeros of the denominator and thus the vertical asymptotes of the function. \(x-1=0\) gives \(x=1\), and \(x+2=0\) gives \(x=-2\).