Chapter 8.4, Problem 56PPSE

Solution

To find: The vertical asymptotes and holes of the graph of the rational function $y=\frac{\left(x-4\right)\left(x+5\right)}{\left(x+3\right)\left(x-4\right)}$ .

The vertical asymptote is $x=-3$ and the hole is $x=4$ of the rational function.

Given:

The rational function is $y=\frac{\left(x-4\right)\left(x+5\right)}{\left(x+3\right)\left(x-4\right)}$ .

Concept:

$\left(a\right)$ Vertical asymptotes of a rational function: The vertical asymptotes of a rational function can be evaluated by reducing the rational function first and equate the denominator equal to zero. If the denominator is already in the reduced form, then simply equate it to zero to find the vertical asymptotes of the function.

$\left(b\right)$ Holes of a rational function: The holes of a rational function can be evaluated by reducing the rational function first and look for common factors in numerator and denominator and equate the common term in numerator and denominator equal to zero to get the holes of the rational function. And, if the rational function is already in reduced form, then there are no holes in the rational function.

Calculation:

The rational function is $y=\frac{\left(x-4\right)\left(x+5\right)}{\left(x+3\right)\left(x-4\right)}$ and the reduce form of rational function is written as $y=\frac{x+5}{x+3}$ by cancelling the common $\left(x-4\right)$ term in numerator and denominator. Now the reduced form of the rational function is $y=\frac{x+5}{x+3}$ . Therefore, using the clause $\left(a\right)$ the vertical asymptote of the rational function can be evaluated by equating the denominator of reduced form of rational function $y=\frac{x+5}{x+3}$ equal to zero. Therefore,

  $\begin{array}{c}y=\frac{\left(x-4\right)\left(x+5\right)}{\left(x+3\right)\left(x-4\right)}\mathrm{....}\left(\text{Given}\right)\mathrm{....}\left(1\right)\\ y=\frac{x+5}{x+3}\mathrm{....}\left(\text{Reduced form}\right)\\ x+3=0\mathrm{....}\left(\begin{array}{l}\text{Set denominator}\\ \text{ equal to zero}\end{array}\right)\\ x=-3\end{array}$

The vertical asymptote is $x=-3$ .

From $\left(1\right)$ , the rational function is $y=\frac{\left(x-4\right)\left(x+5\right)}{\left(x+3\right)\left(x-4\right)}$ where common factor is $\left(x-4\right)$ . Therefore, using the clause $\left(b\right)$ equate $\left(x-4\right)$ equal to zero to get the holes of the rational function.

  $\begin{array}{c}x-4=0\mathrm{....}\left(\begin{array}{l}\text{Equate common }\\ \text{term equal to }0\end{array}\right)\\ x=4\mathrm{....}\left(\begin{array}{l}\text{ which is the hole}\\ \text{of rational funciton}\end{array}\right)\end{array}$

The hole of the rational function is $x=4$ .

Conclusion:

The vertical asymptote is $x=-3$ and the hole is $x=4$ of the rational function.