True or False. The graph of a rational function may intersect a horizontal asymptote.

The Graph Of A Rational Function May Intersect A Horizontal Asymptote.

This article aims to determine whether the given statement is true or false. The statement is, “The graph of a rational function may intersect a horizontal asymptote.” This article uses the concept of horizontal asymptote of the rational function.

A horizontal asymptote is horizontal line that is not part of the graph of a function but leads it for $ x $ values “far” right and “far” left. The graph may intersect it, but eventually, for large enough or small enough values of $ x $, graph would get closer and closer to asymptote without touching it. Horizontal asymptote is a special case of an oblique asymptote.

Horizontal asymptote of rational function can be find by looking at degrees of the numerator and denominator.

If $ N $ is the degree in the numerator and $ D, $ is the degree in the denominator.

-$ N < D $, then the horizontal asymptote is $ y = 0$.

-$ N = D $, then the horizontal asymptote is $ y  = ratio\: of\: leading\: coefficients $.

-$ N > D $, then there is no horizontal asymptote.

Expert Answer

The statement is true. It is possible that graph of a rational function can cross a horizontal asymptote.

Horizontal asymptote of a rational function can find by observing at the degrees of the numerator and denominator.

-The degree of the numerator is less than the degree of the denominator: horizontal asymptote at

-$ y = 0 $

-The degree of the numerator is greater than the degree of the denominator by one: no horizontal asymptote; oblique asymptote.

-The degree of the numerator is equal to the degree of the denominator: the horizontal asymptote in the ratio of the leading coefficients.

Numerical Result

The statement is true. It is possible that the graph of a rational function can cross a horizontal asymptote.

Example

True or False: Graph of a rational function $ R $ never crosses a vertical asymptote. True or False: Graph of a rational function $ R $ never crosses a horizontal asymptote. True or False: Graph of a rational function $ R $ never crosses an oblique asymptote.

Solution

All statements are true.

An asymptote is a line along which the values of a function approaches but never reach, such that one or both of the $ x $ or $ y $ coordinates tend to positive or negative infinity. Therefore, the graph of a rational function $ R $ never intersects any of its asymptotes.

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