Give an example of a function with no asymptotes at all

Question

Give an example of a function with no asymptotes at all.

 

The book gives x² as an answer, but how does that work? x² can't produce a negative value so isn't the asypmtote at 0-ish?

Answer 1

We've learned that the graphs of polynomials are smooth & continuous. They have no asymptotes of any kind.

 

Rational algebraic functions (having numerator a polynomial & denominator another polynomial) can have asymptotes; vertical asymptotes come about from denominator factors that could be zero. Horizontal asymptotes may also exist (when degrees of numerator & denominator are equal, or degree of denominator exceeds degree of numerator).

 

Oblique (slant) asymptotes come about when degree of numerator is exactly 1 more than that of the denominator.

 

So the stated parabola y = x squared qualifies as a "no asymptotes" function graph.

Answer 2

An asymptote is a number (limiting value) that a function approaches as its' input values approach positive or negative infinity. A function can have both horizontal and vertical asymptotes, one or the other, or no asymptote.

 

y=x³ is an example of a function that does not approach any limiting value either horizontally or vertically, and, therefore, has no asymptotes. 

Answer 3

An example of a function with no asymptotes is f(x)=(√x). It has no asymptotes because it is continuous on its domain, which means there are no holes or jumps. Having an asymptote just means that x≠n where n is a point on the function.

 

An example of a function with an asymptote would be (x+1)⁄(x-1)² because x≠1.