Chapter 7.3, Problem 66E

a.

To determine

the relationship between the vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition.

Answer to Problem 66E

The vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition are same.

Explanation of Solution

Given information:

The function is given as $y=\frac{x-12}{x\left(x-4\right)}=\frac{3}{x}-\frac{2}{x-4}$

Concept used:

To find vertical asymptotes of a rational function, simply equate the denominator of rational function to 0 and then solve the equation.

Calculation: Finding the asymptotes of rational function $y=\frac{x-12}{x\left(x-4\right)}$

Here denominator is $x\left(x-4\right)$

So

  $\begin{array}{l}x\left(x-4\right)=0\\ \Rightarrow x=0\text{or}x=4\end{array}$

Asymptotes of rational function are 0 and 4.

Finding the asymptotes of a partial fraction $\frac{3}{x}$

Here the denominator is x , so

  $x=0$

Finding the asymptotes of a partial fraction $\frac{2}{x-4}$

Here the denominator is $x-4$ , so

  $\begin{array}{l}x-4=0\\ x=4\end{array}$

So asymptotes of partial fraction are 0 and 4

Conclusion: The vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition are same i.e. 0 and 4

  

b.

To determine

the relationship between the vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition.

Answer to Problem 66E

The vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition are same.

Explanation of Solution

Given information: The function is given as $y=\frac{2\left(4x-3\right)}{x^2-9}=\frac{3}{x-3}+\frac{5}{x+3}$

Concept used: To find vertical asymptotes of a rational function, simply equate the denominator of rational function to 0 and then solve the equation.

Calculation: Finding the asymptotes of rational function $y=\frac{2\left(4x-3\right)}{x^2-9}$

Here denominator is $x^2-9$

So

  $\begin{array}{l}{x}^2-9=0\\ \left(x-3\right)\left(x+3\right)=0\\ x=3\text{and}x=-3\end{array}$

Asymptotes of rational function are 3 and -3.

Finding the asymptotes of a partial fraction $\frac{3}{x-3}$

Here the denominator is $x-3$

So

  $\begin{array}{l}x-3=0\\ x=3\end{array}$

Finding the asymptotes of a partial fraction $\frac{5}{x+3}$

Here the denominator is $x+3$ , so

  $\begin{array}{l}x+3=0\\ x=-3\end{array}$

So asymptotes of partial fraction are 3 and -3

Conclusion: The vertical asymptotes of the graph of rational function and vertical asymptotes of the graphs representing the partial fractions of the decomposition are same i.e. 3 and -3