Chapter 7: Exercises with Solutions

# Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 7.1 Introducing Rational Functions 615 Version: Fall 2007 7.1 Exercises In Exercises 1 - 14 , perform each of the following tasks for the given rational func- tion. i. Set up a coordinate system on a sheet of graph paper. Label and scale each axis. ii. Use geometric transformations as in Examples 10, 12, and 13 to draw the graphs of each of the following ra- tional functions. Draw the vertical and horizontal asymptotes as dashed lines and label each with its equa- tion. You may use your calculator to check your solution, but you should be able to draw the rational function without the use of a calculator. iii. Use set-builder notation to describe the domain and range of the given rational function. 1. f ( x ) = - 2 /x 2. f ( x ) = 3 /x 3. f ( x ) = 1 / ( x - 4) 4. f ( x ) = 1 / ( x + 3) 5. f ( x ) = 2 / ( x - 5) 6. f ( x ) = - 3 / ( x + 6) 7. f ( x ) = 1 /x - 2 8. f ( x ) = - 1 /x + 4 9. f ( x ) = - 2 /x - 5 10. f ( x ) = 3 /x - 5 11. f ( x ) = 1 / ( x - 2) - 3 Copyrighted material. See: 1 12. f ( x ) = - 1 / ( x + 1) + 5 13. f ( x ) = - 2 / ( x - 3) - 4 14. f ( x ) = 3 / ( x + 5) - 2 In Exercises 15 - 22 , find all vertical as- ymptotes, if any, of the graph of the given function. 15. f ( x ) = - 5 x + 1 - 3 16. f ( x ) = 6 x + 8 + 2 17. f ( x ) = - 9 x + 2 - 6 18. f ( x ) = - 8 x - 4 - 5 19. f ( x ) = 2 x + 5 + 1 20. f ( x ) = - 3 x + 9 + 2 21. f ( x ) = 7 x + 8 - 9 22. f ( x ) = 6 x - 5 - 8 In Exercises 23 - 30 , find all horizontal asymptotes, if any, of the graph of the given function. 23. f ( x ) = 5 x + 7 + 9 24. f ( x ) = - 8 x + 7 - 4

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616 Chapter 7 Rational Functions Version: Fall 2007 25. f ( x ) = 8 x + 5 - 1 26. f ( x ) = - 2 x + 3 + 8 27. f ( x ) = 7 x + 1 - 9 28. f ( x ) = - 2 x - 1 + 5 29. f ( x ) = 5 x + 2 - 4 30. f ( x ) = - 6 x - 1 - 2 In Exercises 31 - 38 , state the domain of the given rational function using set- builder notation. 31. f ( x ) = 4 x + 5 + 5 32. f ( x ) = - 7 x - 6 + 1 33. f ( x ) = 6 x - 5 + 1 34. f ( x ) = - 5 x - 3 - 9 35. f ( x ) = 1 x + 7 + 2 36. f ( x ) = - 2 x - 5 + 4 37. f ( x ) = - 4 x + 2 + 2 38. f ( x ) = 2 x + 6 + 9 In Exercises 39 - 46 , find the range of the given function, and express your an- swer in set notation. 39. f ( x ) = 2 x - 3 + 8 40. f ( x ) = 4 x - 3 + 5 41. f ( x ) = - 5 x - 8 - 5 42. f ( x ) = - 2 x + 1 + 6 43. f ( x ) = 7 x + 7 + 5 44. f ( x ) = - 8 x + 3 + 9 45. f ( x ) = 4 x + 3 - 2 46. f ( x ) = - 5 x - 4 + 9
Section 7.1 Introducing Rational Functions Version: Fall 2007 7.1 Solutions 1. Start with the graph of y = 1 /x in (a). Multiplying by 2 produces the equation y = 2 /x and stretches the graph vertically by a factor of 2 as shown in (b). Multiplying by - 1 produces the equation y = - 2 /x and reflects the graph of y = 2 /x in (b) over the x -axis to produce the graph of y = - 2 /x in (c). Projecting all the points of the graph in (c) onto the x -axis provides the domain: D = { x : x = 0 } . Projecting all the points on the graph in (c) onto the y -axis produces the range: R = { y : y = 0 } . x 10 y 10 y =0 x =0 x 10 y 10 y =0 x =0 x 10 y 10 y =0 x =0 (a) y = 1 /x . (b) y = 2 /x . (c) y = - 2 /x . 3. Start with the graph of y = 1 /x in (a). Replacing x with x - 4 produces the equation y = 1 / ( x - 4) and slides the graph of y = 1 /x four units to the right to produce the graph of y = 1 / ( x - 4) in (b) Projecting all the points of the graph in (b) onto the x -axis provides the domain: D = { x : x = 4 } . Projecting all the points on the graph in (b) onto the y -axis produces the range: R = { y : y = 0 } .

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