Exercise 11 Page 710

Let's recall some information about rational functions and asymptotes. Then we will find the asymptotes of the function from the exercise.

Rational Function

A rational function is a function that contains a rational expression. That is, any function that can be written as the quotient of two polynomial functions p(x) and q(x). f(x) = p(x)/q(x), where q(x) ≠ 0 For any values of x where q(x) = 0, the rational function is undefined. One example of a rational function is the reciprocal function. f(x) = 1/x This function has two asymptotes, the x-axis and the y-axis. The applet shows the graph of the reciprocal function and some other rational functions.

Asymptotes

A line is an asymptote of a graph if the graph gets closer to the line as x or y gets larger in absolute value. For example, the graph of the rational function f(x) = 1x has two asymptotes — the x-axis and the y-axis.

Analyzing the diagram, the following can be observed.

• As x approaches infinity and as x approaches negative infinity, the value of the function approaches 0. Therefore, y = 0 or the x-axis is a horizontal asymptote of the graph of f.
• As x approaches 0, the value of the function approaches either positive or negative infinity. Therefore, x = 0 or the y-axis is a vertical asymptote of the graph of f.

In the coordinate plane below, the asymptotes for three different graphs are shown. The applet demonstrates that asymptotes can be not only vertical and horizontal, but also oblique.

Finding the Asymptotes

To find the asymptotes of the graph of the given function, let's rewrite the given function so that it is in the following form. h(x)= ax- b+ c In this form, the vertical asymptote is x= b and the horizontal asymptote is y= c. Now that the function is written in the correct form, we can see that the vertical asymptote is x= 3 and the horizontal asymptote is y= 0.