Intermediate level Mathematics/Rational functions
Rational function is "any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials".
A function is called a rational function if and only if it can be written in the form
where and are polynomials in and is not the zero polynomial. The domain of a function|domain of is the set of all points for which the denominator is not zero.
However, if and have a non constant polynomial greatest common divisor , then setting and produces a rational function
which may have a larger domain than , and is equal to on the domain of It is a common usage to identify and , that is to extend "by continuity" the domain of to that of Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions A(x)/B(x) and C(x)/D(x) are considered equivalent if A(x)D(x)=B(x)C(x). In this case is equivalent to .
The rational function is not defined at . It is asymptotic to as x approaches infinity.
The rational function is defined for all real numbers, but not for all complex numbers, since if x were a square root of (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero: , which is undefined.
A constant function such as f(x) = π is a rational function since constants are polynomials. Note that the function itself is rational, even though the value (mathematics)|value of f(x) is irrational for all x.
Every polynomial function is a rational function with . A function that cannot be written in this form, such as , is not a rational function. The adjective "irrational" is not generally used for functions.
The rational function is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.
Sketch a graph of a rational function
(1)Let's sketch the graph of .
First, we must avoid because anything can not be divided by 0. Thus x should not be 0 in the equation. Now we just plug in some values of x. The result is as follows:
As x get large the function itself gets smaller and smaller. Here is the graph of .
Paul's Online Math Notes"Rational Functions"