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Chapter 3: Problem 129

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Short Answer

Expert verified
The statement is True. No changes are needed.

Step by step solution

01

Understand the Concept of Vertical Asymptotes

A vertical asymptote is a vertical line \(x = a\) which separates the graph of a function into two halves. As \(x\) approaches \(a\), the value of the function tends to \(\infty\) or \(-\infty\).
02

Understand the Behavior of a Rational Function posessing Vertical Asymptotes

In relation with a rational function, a vertical asymptote occurs when the denominator is equal to zero. This makes the function undefined at that point. Thus, the graph of the function approaches the vertical asymptote but will never cross or touch it.
03

Assessing the Statement

Based on the understanding of vertical asymptotes and their interaction with rational functions, we can now examine the statement: 'The graph of a rational function can never cross a vertical asymptote.' This is indeed true. There is no change necessary.

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Most popular questions from this chapter

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if there is one, of the function's graph.

This will help you prepare for the material covered in the next section. $$\text { Solve: } x^{3}+x^{2}-4 x+4$$

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Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{x^{2}-3 x+2}{x^{2}-2 x-3}>0 $$
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