Problem 129 - College Algebra

Question

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.

Step-by-Step Solution

Answer

The statement is True. No changes are needed.

Step 1: 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 \(-\infty\).

Step 2: 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.

Step 3: 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.