Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

Step-by-Step Solution

Answer

False. A rational function can cross its horizontal or oblique asymptotes.

Step 1: Understanding Rational Functions and Asymptotes

A rational function is expressed as a ratio of two polynomials. Asymptotes are lines that a function approaches as it heads toward infinity or negative infinity. Understanding this is vital to solving the problem at hand.

Step 2: Understanding Asymptote Crossing

The function will never meet its vertical asymptotes because these are values for which the function is undefined. However, it is possible for the function to cross its horizontal or oblique asymptotes. This is because these asymptotes describe the behavior of the function as it heads towards positive or negative infinity, not at specific finite points.

Step 3: Verdict on the Statement

The statement 'The graph of a rational function can never cross one of its asymptotes.' is not entirely accurate. It would be correct if it specified 'vertical asymptotes'. However, since it does not, and rational functions can cross their horizontal or oblique asymptotes, the overall statement is false.