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# Mth141s41 - Section 4.1 Rational Functions and Graphs...

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Section 4.1 - Rational Functions and Graphs Rational functions = ????? Recall that a rational number is a number that can be written as the ratio of two integers. A rational expression was the ratio of two polynomials. For ex. 2 3 5 4 x x x - + + . So it makes sense to define a rational function as the ratio of two polynomial functions. DEFINITION: A rational function is a function f that is the quotient of two polynomials, that is , ( ) ( ) ( ) p x f x q x = , where p(x) and q(x) are polynomials and q(x) is not equal to zero. Ex. The following are some examples of rational functions: 1 ( ) f x x = 1 ( ) 1 x g x x + = - 3 2 3 5 ( ) 4 x x h x x + = - An important idea is that the domain of a rational function is all real numbers, except those that make the denominator zero. So the domains of the functions above are: f(x) x 0 g(x) x 1 and h(x) x 2 and x -2 or using interval notation : f(x) ( ,1) (1, ) -∞ +∞ U g(x) ( ,1) (1, ) -∞ +∞ U and h(x) ( 29 , 2) ( 2,2 (2, ) -∞ - - +∞ U U HORIZONTAL ASYMPTOTES: Note the graph of 1 ( ) f x x =

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As x gets very large in value, x
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Mth141s41 - Section 4.1 Rational Functions and Graphs...

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