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Question 538978: I need to be able to create a rational function with the following qualities:
has no vertical asymptote
has no horizontal asymptote
has no x-intercept
Thanks for any help you can give me!
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A rational function would be expected to involve a quotient of polynomials
Vertical asymptotes happen (often, but not always) where a denominator is zero as for  at x=0.
Exception: A denominator can be zero without a vertical asymptote if the function could be simplified, with domain restrictions, to one whose denominator is never zero, as in  , which can be simplified to y=1 for all x except x=0, which is not in the domain of the original quotient.
Horizontal asymptotes happen when for very large |x| the function can be approximated by a number as for  or  .
That happens with a quotient of polynomials when the degree of the numerator is not greater than the degree of the denominator.
The x-intercept is where the function equals zero. That happens for a quotient of polynomials when the numerator is zero.
So, if your function is a quotient of polynomials, you could use a denominator that is never zero, and a numerator that is never zero and has a greater degree.
 is an example. Can you think of another one?
HINT:
Polynomials of even degree can stay on one side of the x axis and never be zero. For example, you know that  or  or  or ... touch the x-axis only at x=0. If you add a positive number to get something like  or  you know that sum will be always positive, never zero.
Polynomials of odd degree are obligated to cross the x-axis because they go from -infinity at one end of the x-axis to +infinity at the other end.
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