Chapter EP, Problem 9.4.3EP

To determine

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

Answer to Problem 9.4.3EP

  $x=-1$ and $x=8$ are vertical asymptotes of the given rational function.

Explanation of Solution

Given: $f\left(x\right)=\frac{5}{(x+1)(x-8)}$

$f\left(x\right)=\frac{5}{(x+1)(x-8)}$ be the given rational function and it has to find any vertical asymptotes and any holes for the value of $x$ .

Now $f\left(x\right)=\frac{5}{(x+1)(x-8)}$ is given equation and $x+1$ and $x-8$ are its denominator.

For vertical asymptotes $x+1=0$

  $x+1=0$

  $x=-1$

Here in solution there is no factor that cancels in the numerator.

And $x=-1$ is the identical factor in the numerator that is vertical asymptote.

For vertical asymptotes $x-8=0$

  $x-8=0$

  $x=8$

Here in solution there is no factor that cancels in the numerator.

And $x=8$ is the identical factor in the numerator that is vertical asymptote

Now graph of the function $f\left(x\right)=\frac{5}{(x+1)(x-8)}$ is

  

Hence, $x=-1$ and $x=8$ are vertical asymptotes of the given rational function.