Question

Show all worked identify the asymptotes and state the end behavior of the function F(x)=6x over x-36

Answer 1

Solution

Asymptote:

Vertical Asymptote

- The vertical asymptotes of a rational function are determined by the denominator expression.

- The expression given is:

$f(x)=\frac{6x}{x-36}$

- The denominator of (x- 36) determines the asymptote line.

- The vertical asymptote defines where the rational function isundefined. Iin order for a rational function to be undefined, its denominator must be zero.

- Thus, we can say:

$\begin{gathered} x-36=0 \\ Add\text{ 36 to both sides} \\ \\ \therefore x=36 \end{gathered}$

- Thus, the vertical asymptote is

$x=36$

Horizontal Asymptote:

- The horizontal asymptote exists in two cases:

1. When the highest degree of the numerator is less han the degree of the demnominator. In this case, the horizontal asymptote is y = 0

2. When the highest degee sof the numerator and tdenominator are the same. In this case, the horizontal asymptote is

$\begin{gathered} y=\frac{N}{D} \\ where, \\ N=\text{ Coefficient of the highest degree of the numerator} \\ D=\text{ Coefficient of the highest degree of the denominator} \end{gathered}$

- For our question, we can see that the highest degrees of the numerator and denominator are the same. Thus, we have the Horizontal Asymptote to be:

$y=\frac{6}{1}=6$

End behavior:

- The end behavior is examining the y-values of the function as x tendsto negative and positive infinity.

- Thus, we have that:

$\begin{gathered} f(x)=\frac{6x}{x-36} \\ \\ \text{ Divide top and bottom by }x \\ f(x)=\frac{6x}{x-36}\times\frac{x}{x} \\ \\ f(x)=\frac{\frac{6x}{x}}{\frac{x-36}{x}}=\frac{6}{1-\frac{36}{x}} \\ \\ As\text{ }x\to-\infty \\ f(-\infty)=\frac{6}{1-\frac{36}{-\infty}}=\frac{6}{1+\frac{36}{\infty}}=\frac{6}{1+0}=6 \\ \\ \text{ Thus, we can say: }x\to-\infty,f(x)\to6 \\ \\ Also, \\ As\text{ }x\to\infty \\ f(\infty)=\frac{6}{1-\frac{36}{\infty}}=\frac{6}{1-0}=6 \\ \\ \text{ Thus, we can also say: }x\to\infty,f(x)\to6 \end{gathered}$

Final Answers

Asymptotes:

$\begin{gathered} \text{ Vertical:} \\ x=36 \\ \\ \text{ Horizontal:} \\ y=6 \end{gathered}$

End behavior:

$\begin{gathered} As\text{ }x\to-\infty,f(x)\to6 \\ \\ As\text{ }x\to\infty,f(x)\to6 \end{gathered}$