Rational Function

Question

If possible, write the equation of a rational function with all of the following properties:

Horizontal asymptote of y = 5,

Vertical asymptotes of x = 3 and x = -8

Continuous for all x not equal to 3 or -8.

Answer 1

In general, the rational function will be of the form

R(x) = A P(x)/Q(x)

where A is a constant, and P(x), Q(x) are polynomials with leading coefficient 1.

1) Horizontal asymptote of y = 5 implies that

A = 5, and P and Q must be of the same degree.

This is the reason:

If degree(P) < degree(Q) then R(x) -> 0 when x -> ∞ or -∞

If degree(P) > degree(Q) then R(x) -> ∞ or -∞ when x -> ∞ or -∞

If degree(P) = degree(Q) then P(x)/Q(x) -> 1 when x -> ∞ or -∞

2) Vertical asymptotes of x = 3 and x = -8 implies that

Q(x) must have the roots 3 and -8, and P(x) must not have those root.

If Q(x) did not those roots that R(3) and R(-8) would be defined.

If P(x) had a root 3 or -8, then R(3) or R(-8) would be a hole, not a vertical asymptote.

3) Continuous for all x not equal to 3 or -8 implies that

Q(x) cannot have any other roots besides those two.

The simplest rational function satisfying those constraints is

R(x) = 5x²/(x-3)(x+8)