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Rational Functions
Significance "Rational function" is the name given to a function
which can be represented as the quotient of polynomials, just as a
rational number is a number which can be expressed as a quotient of
whole numbers. Rational functions supply important examples and occur
naturally in many contexts. All polynomials are rational functions.
Standard
Notation
The typical rational function has the form p(x)/q(x) where p and q are polynomials. p(x) is called the numerator and q(x) is called the denominator.
Example: In the rational function
| x2 -
4 |
| g(x) = | ------------------ |
| x2 - 5x + 6
|
the numerator is x2 - 4 and the denominator is x22 - 5x + 6. A polynomial is a rational functions with denominator 1.
The Domain of a Rational Function
The domain of the rational function p(x)/q(x) consists of all points x where q(x) is non-zero. This domain really depends on the way in which p(x) and q(x) are chosen. For example, the function g(x) above can be written as | (x - 2)(x + 2) |
| g(x) = |
|
| (x - 3)( x - 2)
|
which simplifies to
simplifies to (x+2)/(x-3), but the x = 2 is NOT in the domain of g(x) whereas it IS in the domain of (x+2)/(x-3).
Other Quotients
Functions which are quotients of functions other than polynomials are
not called rational functions, but the same considerations apply: The
domain of a quotient includes only points where - both the
numerator and the denominator are defined, and
- the denominator is
not 0.
Graphing Rational Functions
| It is often better to use algebra and differential calculus to make a sketch which illustrates the important features, rather than using technology. "Curve sketching" is one of the topics covered in Stage 7 of CalculusQuestTM. |
The graphs of rational functions can be very complicated sporting bumps, holes, and asymptotes. Most graphing calculators do a lousy job of graphing
rational functions unless you know how to tell them to do a good job. This involves adjusting window parameters and graphing the function in pieces which display all the important features and avoid numerical difficulties. Here are some examples of different raw views of the same rational function f(x) = (x3 - 2 x2 + 1)/(x2 - 4), illustrating how different features show up and are obscured. The vertical lines and spikes, put in by the graphing routine, are numerical artifacts caused asymptotes. They are not really part of the graph!
As a second example, the graph of the function g(x) above is identical to the graph of the function (x + 2)/(x - 3) except that it has a missing point (hole) at x = 2 which is not in the domain of g(x). You are unlikely to get your calculator to show this feature -- how do you accurately "draw" a missing point which, after all, has no length or width?
Asymptotes
An asymptote for a function f(x) is a straight line which is approached but never reached by f(x). Rational functions exhibit three types of asymptotes: - Vertical asymptotes. These are vertical lines near which the function f(x) becomes infinite. If the denominator of a rational function has more factors of (x - a) than the numerator, then the rational function will have a vertical asymptote at x = a.
- Horizontal Asymptotes. A horizontal asymptote is a line y = c such that the values of f(x) get increasingly close to the number c as x gets large in either the positive or negative direction. Rational functions have horizontal asymptotes when the degree of the numerator is the same as the degree of the denominator.
- Oblique Asymptotes An oblique asymptote is an asymptote of the form y = a x + b with a non-zero. Rational functions have obliques asymptotes if the degree of the numerator is one more than the degree of the denominator. The function g(x) above has an oblique asymptote, namely the line y = x. This is reflected in the last view of the graph of g(x) on a calculator.
Examples are on the next page, and lots more are in Stage 3.
Calculations with Rational Functions
If f(x) = p(x)/q(x) is a rational function, the equation f(x) = b is the same as the equation p(x) = b q(x) or p(x) - b q(x) = 0. The function p(x) - b q(x) is a polynomial, so solving an equation involving a rational function reduces to finding roots of a polynomial.
Examples and Calculations are on the next page.
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