Rational Functions
Algebraic Representation
f(x) = P(x) / Q(x) |
A rational function is a quotient (ratio) of two polynomials P and Q . To understand the behavior of rational functions, all you need to understand are ...quotients and polynomials!
Quotients (a.k.a. fractions) are simple enough: Just remember that the size of a quotient is proportional to the size of the numerator (P) and inversely proportional to the size of the denominator (Q).
We use what we know about polynomials to understand the numerator and the denominator individually, then use what we know about fractions to understand the rational function as a whole.
Consider an example:
We factor both the numerator and the denominator (when possible) to get a look at the roots of P and Q . We also like to expand both polynomials to see the terms of highest degree – the dominant terms. These are all we need to examine the short and long term behaviors, respectively, of the entire function.
When the numerator is zero, the entire quotient is zero (provided Q isn't zero at the same place): The roots of P are the roots of f . In this case, P and f have roots at
When the denominator is zero, the quotient is undefined: The roots of Q are the singularities of f . In this case, f has singularities at
When the input of f is near a singularity – when the denominator of the quotient is very small – f "blows up" in inverse proportion. Whether or not f travels through large positive or large negative values near a singularity depends on the relative signs of P and Q in that neighborhood. For example, to the left of
In the long term (far from the origin), both P and Q behave like their dominant terms. The quotient, therefore, behaves like the ratio of these terms. Suppose P's largest power is
There are three possibilities:
In the example, the asymptote is
This tells us that:
Notice that, in the fraction being subtracted from x , the denominator is always positive in the long term
The key algebraic properties of rational functions are simply the properties of fractions. You must know how changes in the numerator and the denominator affect the entire ratio. To understand the behaviors of the numerator and the denominator individually, of course, you must be familiar with the algebra of polynomials.