Question

write a rational function that satisfies the given conditions.

1. write a function that has 3 vertical asymptotes at x = 3, x = 5, and x = - 2.

Explanation:

Step1: Recall the property of vertical asymptotes

A rational function $y = \frac{f(x)}{g(x)}$ has a vertical asymptote at $x = a$ when $g(a)=0$.

Step2: Construct the denominator

Since the vertical asymptotes are at $x = 3$, $x = 5$, and $x=-2$, the denominator $g(x)$ of the rational - function can be $(x - 3)(x - 5)(x + 2)$.

Step3: Construct the rational function

We can choose the numerator $f(x)$ to be a non - zero polynomial. For simplicity, let $f(x)=1$. Then the rational function is $y=\frac{1}{(x - 3)(x - 5)(x + 2)}$.

Answer:

$y=\frac{1}{(x - 3)(x - 5)(x + 2)}$