Review Exercises AN31 11. Step 1: = y x4 Step 2: x-intercepts: − − 3, 1, 1; y-intercept: 3 Step 3: − − 3, 1: multiplicity 1; crosses; 1: multiplicity 2; touches Step 4: 80 24 220 4 Step 5: ( ) ( ) − − − 2.28, 9.91 , 0.22, 3.23 , ( ) 1, 0 Step 6: y 80 x (1, 0) 5 (0, 3) (2, 15) (22.28, 29.91) (23, 0) (21, 0) (20.22, 3.23) (24, 75) Step 7: Range: [ ) − ∞ 9.91, Step 8: Increasing on [ ] − − 2.28, 0.22 and [ )∞1, Decreasing on ( ] −∞ −, 2.28 and [ ] −0.22, 1 12. = R 10; g is not a factor of f . 13. = R 0; g is a factor of f . 14. ( ) = f 4 47,105 15. 4, 2, or 0 positive; 2 or 0 negative 16. 1 positive; 2 or 0 negative 17. ± ± ± ± ± ± ± ± ± 1, 3, 1 2 , 3 2 , 1 3 , 1 4 , 3 4 , 1 6 , 1 12 18. ( ) ( )( )( ) − = + − − f x x x x 2, 1, 4; 2 1 4 19. 1 2 , multiplicity ( ) ( ) ( ) − = − + f x x x 2; 2; 4 1 2 2 2 20. 2, multiplicity 2; ( ) ( ) ( ) = − + f x x x 2 5 2 2 21. { } −3, 2 22. { } − − − 3, 1, 1 2 , 1 23. −2 and 3 20 23 210 4 25. ( ) ( ) = − = f f 0 1; 1 1 26. ( ) ( ) = − = f f 0 1; 1 1 27. 1.52 28. 0.93 29. ( ) − = − + − i f x x x x 4 ; 14 65 102 3 2 30. ( ) − − = − + − + i i f x x x x x , 1 ; 2 3 2 2 4 3 2 31. ( ) ( )( )( ) − = + − − f x x x x 2, 1, 4; 2 1 4 32. ( ) ( ) ( ) ( ) − = + − f x x x 2, 1 2 multiplicity 2 ; 4 2 1 2 2 24. −5 and 5 40 25 210 5 33. 2 (multiplicity 2), ( )( ) ( ) ( ) − = + − − i i f x x i x i x 5, 5; 5 5 2 2 34. ( ) ( )( ) − − = + − + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ i i f x x x x i x i 3, 2, 2 2 , 2 2 ; 2 3 2 2 2 2 2 35. Domain: { } ≠ − ≠ x x x 3, 3 : horizontal asymptote: = y 0; vertical asymptotes: = − = x x 3, 3 36. Domain: { } ≠ − x x 2 ; horizontal asymptote: = y 1; vertical asymptote: = − x 2 37. Step 1: ( ) ( ) = − R x x x 2 3 ; domain: { } ≠ x x 0 Step 2: R is in lowest terms Step 3: no y-intercept; x-intercept: 3 Step 4: R is in lowest terms; vertical asymptote: = x 0 Step 5: Horizontal asymptote: = y 2; not intersected Step 6: 9 25 25 6 Step 7: y 10 x 210 (1, 24) (3, 0) (22, 5) y 5 2 x 5 0 4, 1 2 38. Step 1: Domain: { } ≠ ≠ x x x 0, 2 Step 2: H is in lowest terms Step 3: no y-intercept; x-intercept: −2 Step 4: H is in lowest terms; vertical asymptote: = = x x 0, 2 Step 5: Horizontal asymptote: = y 0; intersected at ( ) −2, 0 Step 6: 5 25 25 5 Step 7: y 5 x 5 (1, 23) (22, 0) 23, 2 1 15 3, 5 3 21, 1 3 x 5 2 x 5 0 39. Step 1: ( ) ( )( ) ( )( ) = + − − + R x x x x x 3 2 3 2 ; domain: { } ≠ − ≠ x x x 2, 3 Step 2: R is in lowest terms Step 3: y-intercept: 1; x-intercepts: −3, 2 Step 4: R is in lowest terms; vertical asymptotes: = − = x x 2, 3 Step 5: Horizontal asymptote: = y 1; intersected at ( ) 0, 1 Step 6: 5 25 25 7 Step 7: y x 5 5 (2, 0) (0, 1) (23, 0) 4, 7 3 ,2 11 9 5 2 24, 3 7 2 , 2 9 11 5 2 y 5 1 x 5 22 x 5 3 40. Step 1: ( ) ( )( ) = + − F x x x x 2 2 3 ; domain: { } ≠ − ≠ x x x 2, 2 Step 2: F is in lowest terms Step 3: y-intercept: 0; x-intercept: 0 Step 4: F is in lowest terms; vertical asymptotes: = − = x x 2, 2 Step 5: Oblique asymptote: = y x; intersected at ( ) 0, 0 Step 6: 10 26 210 6 Step 7: y 4 x 10 (0, 0) 3, 27 5 21, 1 3 23, 2 27 5 y 5 x x 5 22 x 5 2 1, 2 1 3
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