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Rational Functions

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A rational function is a function that can be written as the quotient of two polynomials. Any rational function r(x) = , where q(x) is not the zero polynomial. Because by definition a rational function may have a variable in its denominator, the domain and range of rational functions do not usually contain all the real numbers.

There is special symbolism to describe the behavior of a function in certain situations, depending on the behavior of the independent variable. In speaking one might say a function approaches a certain value as x increases, decreases, or approaches a certain value. To mathematically say "approaches," an arrow is used. For example, to say that the function f (x) increases without bound as x increases without bound, one would write f (x)âÜ’âàû as xâÜ’âàû . Or to say the function f decreases without bound as x approaches 0 , you would write f (x)âÜ’ - âàû as xâÜ’ 0 .

Rational functions often have what are called asymptotes. Asymptotes are lines that functions approach but never reach. There are three kinds of asymptotes: vertical, horizontal, and oblique. A vertical asymptote is a line with the equation x = h if f (x)âÜ’±âàû as xâÜ’h from either direction. A horizontal asymptote is a line with the equation y = k if f (x)âÜ’k as xâÜ’±âàû . Oblique asymptotes are linear functions.

Study the graph below of the rational function f (x) = .

Figure %: The graph of f (x) = .
The line x = 0 is a verical asymptote and y = 0 is a horizontal asymptote.

A line x = h is a vertical asymptote of a function f (x) = if p(h)≠ 0 and q(h) = 0 . This is the general form of all vertical asymptotes of rational functions.

Horizontal asymptotes are a little trickier to understand. Let f (x) = . If the degree of p is less than that of q , then y = 0 is a horizontal asymptote of f . If the degree of p is greater than that of q , then f does not have a horizontal asymptote. If p and q have the same degree, then the horizontal asymptote occurs at the line y = , where candd are the leading coefficients of p and q , respectively.