create equations of rational functions with the following characteristics.?

Question

No horizontal and no vertical asymptotes

Answer 1

y=(3x⁴+ 10x³ - 72x² + 38x + 21)/(x² + 4x - 21)

Note that a rational function contains a vertical asymptote iff the denominator contains a factor that the numerator does not, and a horizontal asymptote iff the denominator is of greater or equal degree to the numerator. This function has a denominator of lesser degree than the numerator, and every factor of the denominator is a factor of the numerator, therefore, it will not possess asymptotes.

Answer 2

The only thing I would add to Pascal's answer (I assume he's correct about the factors -- I haven't bothered to check that
x + 7 and x - 3 are factors of the numerator) is that you could use a denominator which has no zeros, e.g. x^2 + 4x + 21, so that there are no vertical asymptotes, and provided the numerator has higher degree than the denominator there will be an asymptote but it isn't horizontal.

Some mathematics teachers use the term "asymptote" even when there's just an isolated point missing from the function, so they would say that x = 3 and x = -7 are in fact asymptotes of Pascal's example.

Answer 3

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Answer 4

Im too busy right now. Try back later.