236 CHAPTER 4 Polynomial and Rational Functions Although − − x x 1 1 2 simplifies to +x 1, it is important to observe that the functions ( ) ( ) = − − = + R x x x f x x 1 1 and 1 2 are not equal, since the domain of R is { } ≠ x x 1 and the domain of f is the set of all real numbers. Notice in Table 10 there is an error message for ( ) = = − − Y R x x x 1 1 , 1 2 but there is no error message for ( ) = = + Y f x x 1. 2 4.5 Properties of Rational Functions Now Work the ‘Are You Prepared?’ problems on page 244. • Rational Expressions (Section A.5, pp. A35–A42) • Polynomial Division (Section A.3, pp. A25–A27) • Graph of ( ) = f x x 1 (Section 1.3, Example 9, pp. 25–26) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) PREPARING FOR THIS SECTION Before getting started, review the following: Ratios of integers are called rational numbers . Similarly, ratios of polynomial functions are called rational functions . Examples of rational functions are ( ) ( ) ( ) = − + + = − = − R x x x x F x x x G x x x 4 1 4 3 1 2 2 3 2 2 4 OBJECTIVES 1 Find the Domain of a Rational Function (p. 236) 2 Find the Vertical Asymptotes of a Rational Function (p. 240) 3 Find a Horizontal or an Oblique Asymptote of a Rational Function (p. 241) Finding the Domain of a Rational Function (a) The domain of ( ) = − + R x x x 2 4 5 2 is the set of all real numbers x except −5; that is, the domain is { } ≠ − x x 5 . (b) The domain of ( ) ( )( ) = − = + − R x x x x 1 4 1 2 2 2 is the set of all real numbers x except −2 and 2; that is, the domain is { } ≠ − ≠ x x x 2, 2 . (c) The domain of ( ) = + R x x x 1 3 2 is the set of all real numbers because + x 1 2 is always greater than or equal to 1 over the set of real numbers. (d) The domain of ( ) = − − R x x x 1 1 2 is the set of all real numbers x except 1; that is, the domain is { } ≠ x x 1 . EXAMPLE 1 1 Find the Domain of a Rational Function DEFINITION Rational Function 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 of R is the set of all real numbers, except those for which the denominator q is 0. Now Work PROBLEM 17
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