Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Find the domain, points of discontinuity, and x- and y- intercepts of each rational function. Determine whether the discontinuities are removable or non-removable.

13. y = 2 x 2 + 5 x 2 − 2 x y equals . fraction 2 , x squared , plus 5 , over x squared , minus 2 x end fraction
14. y = x 2 + 2 x x 2 + 2 y equals . fraction x squared , plus 2 x , over x squared , plus 2 end fraction
15. y = 3 x − 3 x 2 − 1 y equals . fraction 3 x minus 3 , over x squared , minus 1 end fraction
16. y = 6 − 3 x x 2 − 5 x + 6 y equals . fraction 6 minus 3 x , over x squared , minus 5 x plus 6 end fraction

See Problem 2.

Find the vertical asymptotes and holes for the graph of each rational function.

17. y = 3 x + 2 y equals . fraction 3 , over x plus 2 end fraction
18. y = x + 5 x + 5 y equals . fraction x plus 5 , over x plus 5 end fraction
19. y = x + 3 ( 2 x + 3 ) ( x − 1 ) y equals . fraction x plus 3 , over open , 2 x plus 3 , close . open , x minus 1 , close end fraction
20. y = ( x + 3 ) ( x − 2 ) ( x − 2 ) ( x + 1 ) y equals . fraction open , x plus 3 , close . open , x minus 2 , close , over open , x minus 2 , close . open , x plus 1 , close end fraction
21. y = x 2 − 4 x + 2 y equals . fraction x squared , minus 4 , over x plus 2 end fraction
22. y = x + 5 x 2 + 9 y equals . fraction x plus 5 , over x squared , plus 9 end fraction

See Problem 3.

Find the horizontal asymptote of the graph of each rational function.

23. y = 5 x + 6 y equals . fraction 5 , over x plus 6 end fraction
24. y = x + 2 2 x 2 − 4 y equals . fraction x plus 2 , over 2 , x squared , minus 4 end fraction
25. y = x + 1 x + 5 y equals . fraction x plus 1 , over x plus 5 end fraction
26. y = x 2 + 2 2 x 2 − 1 y equals . fraction x squared , plus 2 , over 2 , x squared , minus 1 end fraction
27. y = 5 x 3 + 2 x 2 x 5 − 4 x 3 y equals . fraction 5 , x cubed , plus 2 x , over 2 , x to the fifth , minus 4 , x cubed end fraction
28. y = 3 x − 4 4 x + 1 y equals . fraction 3 x minus 4 , over 4 x plus 1 end fraction

See Problem 4.

Sketch the graph of each rational function.

29. y = x 2 − 4 3 x − 6 y equals . fraction x squared , minus 4 , over 3 x minus 6 end fraction
30. y = 4 x x 3 − 4 x y equals . fraction 4 x , over x cubed , minus 4 x end fraction
31. y = x + 4 x − 4 y equals . fraction x plus 4 , over x minus 4 end fraction
32. y = x ( x + 1 ) x + 1 y equals . fraction x . open , x plus 1 , close , over x plus 1 end fraction
33. y = x + 6 ( x − 2 ) ( x + 3 ) y equals . fraction x plus 6 , over open , x minus 2 , close . open , x plus 3 , close end fraction

    See Problem 5.

34. y = 3 x ( x + 2 ) 2 y equals . fraction 3 x , over open , x plus 2 , close squared end fraction
35. Pharmacology How many milliliters of the 0.5% solution must be added to the 2% solution to get a 0.65% solution? Use the rational function given in Problem 5.

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