SECTION 4.5 Properties of Rational Functions 245 10. True or False The graph of a rational function may intersect a vertical asymptote. 11. If a rational function is proper, then _______ is a horizontal asymptote. 12. True or False If the degree of the numerator of a rational function equals the degree of the denominator, then the rational function has a horizontal asymptote. 13. Multiple Choice If ( ) ( ) ( ) = R x p x q x is a rational function and if p and q have no common factors, then R is . (a) improper (b) proper (c) undefined (d) in lowest terms 14. Multiple Choice Which type of asymptote, when it occurs, describes the behavior of a graph when x is close to some number? (a) vertical (b) horizontal (c) oblique (d) all of these Skill Building In Problems 15–26, find the domain of each rational function. 15. ( ) = − R x x x 4 7 16. ( ) = + R x x x 5 3 2 17. ( ) ( )( ) = − − + H x x x x 4 2 4 2 18. ( ) ( )( ) = + − G x x x 6 3 4 19. ( ) ( ) = − − − F x x x x x 3 1 2 5 12 2 20. ( ) ( ) = − − + − Q x x x x x 1 3 5 2 2 21. ( ) = − R x x x 64 3 22. ( ) = − R x x x 1 4 23. ( ) = + + H x x x x 3 9 2 2 24. ( ) = − + G x x x 3 1 4 25. ( ) ( ) ( ) = − − − R x x x x 3 6 5 4 2 2 26. ( ) ( ) ( ) = − − + + F x x x x 2 4 3 4 4 2 2 In Problems 27–32, use the graph shown to find (a) The domain and range of each function (b) The intercepts, if any (c) Horizontal asymptotes, if any 27. x y –4 4 4 –4 28. x y 23 3 3 23 (0, 2) 29. x y 23 3 3 23 (1, 0) (21, 0) (1, 2) 30. x y 23 3 3 23 (21, 2) (21, 1) (1, 22) 31. x y 23 3 3 23 32. x y 23 3 3 23 In Problems 33–44, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. 33. ( ) = + F x x 2 1 34. ( ) = + Q x x 3 1 2 35. ( ) ( ) = − R x x 1 1 2 36. ( ) = R x x 3 (d) Vertical asymptotes, if any (e) Oblique asymptotes, if any
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