**Unformatted text preview: **Section 3.3
Rational Functions Rational Functions
A rational function is a function of the form R(x) = p(x) / q(x) where p and q are polynomial functions and q is not the zero polynomial. The domain is the set of all real numbers except those for which the denominator, i.e. q, is zero. Examples
What is the domain of 4x / (x 3) x(1 x) / (3x2 + 5x 2) (x 3) / (x4 + 1) Horizontal and Vertical Asymptotes
If as x or as x , the values of the rational function, R(x) approach some fixed number, L, then the line y = L is a horizontal asymptote of the graph of R. If as x approaches some number, c, the values of |R(x)| , then the x = c is a vertical asymptote of the graph of R. Note that a graph my cross a horizontal asymptote. Theorem: Locating Vertical Asymptotes
A rational function R(x) = p(x) / q(x), in lowest terms, will have a vertical asymptote x = r if r is a real zero of the denominator q. Example: 3(x2 x 6) / 4(x2 9) Horizontal Asymptotes
If the degree of the numerator, n, is less than the degree of the denominator, m, then the rational function is said to be proper. If a rational function is proper, the line y = 0 is a horizontal asymptote. If n = m then the line y = L, where L is the ratio of the leading coefficients, will be a horizontal asymptote. Example: (3x2 + x) / (5x2 + 4x 2) Oblique Asymptotes
Let n be the degree of the numerator, and let m be the degree of the denominator, if n=m+1 then the rational function has an oblique asymptote. To find the oblique asymptote you perform the long division and the quotient will be of the form ax + b, and the line y = ax + b will be an oblique asymptote. Note a graph my cross an oblique asymptote. Summary
1. 2. 3. 4. If n < m then R is proper and has a horizontal asymptote of y = 0. If n = m, then the line y = L, where L is the ratio of the leading coefficients. If n = m + 1, perform long division and the quotient, y = ax + b, will be the oblique asymptote. If n > m + 1, the quotient will be of degree 2 or higher and there is no horizontal or oblique asymptote. ...

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