A rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.
In the case of one variable, x , a function is called a rational function if and only if it can be written in the form
f(x)=P(x)Q(x)
where P and Q are polynomial functions in and is not the zero polynomial. The domain of f is the set of all points for which the denominator is not zero, where one assumes that the fraction is written in its lower degree terms, that is, P and Q have several factors of the positive degree. Every polynomial function is a rational function with Q(x)=1. A function that cannot be written in this form (for example, f(x)=sin(x)) is not a rational function (but the adjective "irrational" is not generally used for functions, but only for real numbers). An expression of the form is called a rational expression. The x need not be a variable. In abstract algebra the x is called an indeterminate. A rational equation is an equation in which two rational expressions are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected. The rational function f(x)=x3-2x2(x2-5) is not defined at x2=5⇔x=±√5. The rational function f(x)=x2+2x2+1 is defined for all real numbers, but not for all complex numbers, since if x were the square root of −1(i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: f(i)=i2+2i2+1=-1+2-1+1=10, which is undefined. -
The domain of a rational function is all real values except where the denominator, Q(x)=0.
Roots
The roots,( zeros, solutions, or x-intercepts) of the rational function will be the places where P(x)=0. That is, completely ignore the denominator. Whatever makes the numerator zero will be the roots of the rational function, just like they were the roots of the polynomial function earlier.
If you can write it in factored form, then you can tell whether it will cross or touch the x-axis at each
x-intercept by whether the multiplicity on the factor is odd or even.
Vertical Asymptotes
A vertical asymptote is a vertical line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of Q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.
If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd.
Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.
The graph pictured here is of the function f(x)=x+1(x+4). The zeros of the denominator is
A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not (necessarily) asymptotic in the middle. It is possible for the graph of a function to cross a horizontal asymptote.
Given or more precisely
The location of the horizontal asymptote is determined by looking at the degrees of the numerator (n) and denominator (m).
If , the x-axis, (or ) is the horizontal asymptote.
If , then is the horizontal asymptote. That is, the ratio of the leading coefficients.
If , there is no horizontal asymptote.
Note: if , there is an oblique or slant asymptote.
Sometimes, a factor will appear in the numerator and in the denominator. Let's assume the factor is in the numerator and denominator. Because the factor is in the denominator, will not be in the domain of the function. This means that one of two things can happen. There will either be a vertical asymptote at , or there will be a hole at .
Let's look at what will happen in each of these cases.
There are more factors in the denominator. After dividing out all duplicate factors, the is still in the denominator. Factors in the denominator result in vertical asymptotes. Therefore, there will be a vertical asymptote at .
There are more factors in the numerator. After dividing out all the duplicate factors, the is still in the numerator. Factors in the numerator result in x-intercepts. But, because you can't use , there will be a hole in the graph on the x-axis.
There are equal numbers of factors in the numerator and denominator. After dividing out all the factors (because there are equal amounts), there is no left at all. Because there is no in the denominator, there is no vertical asymptote at x = k. Because there is no in the numerator, there is no x-intercept at . There is just a hole in the graph, someplace other than on the x-axis. To find the exact location, plug in into the reduced function (you can't plug it into the original, it's undefined, there), and see what y-value you get.
