Chapter 9, Problem 31SGR

To determine

The equations of any vertical asymptotes and the values of x for ny holes in the graph of rational function.

Answer to Problem 31SGR

The vertical asymptotes are $x=0$ and $x=-4$ and there is no holes the graph.

Explanation of Solution

Given information:

The given rational function is $f\left(x\right)=\frac{3}{x^2+4x}$ .

Calculation:

If $f\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}$ , $a\left(x\right)$ and $b\left(x\right)$ are polynomial function with no common factor other than 1, and

  then has vertical asymptotes whenever

And if is the factor of both and then there is a point of discontinuity at

Therefore, for the function

  

So the function is undefined for and

So the vertical asymptotes are and and since there is no common factor thus there will be no holes in the graph.

Hence,

The vertical asymptotes are and and there is no holes the graph.