AN12 Answers: Chapter 2 20. (a) Domain: { } ≤ x x 4 or ( ] −∞, 4 Range: { } ≤ y y 3 or ( ] −∞, 3 (b) Increasing on ( ] −∞ −, 2 and [ ] 2, 4 ; Decreasing on [ ] −2, 2 (c) Local maximum value is 1 and occurs at = − x 2. Local minimum value is −1 and occurs at = x 2. (d) Absolute maximum: ( ) = f 4 3 Absolute minimum: none 17. ( )( ) ( ) + = + − − f g x x x x x 2 1 1 ; 2 Domain: { } ≠ ≠ x x x 0, 1 ( )( ) ( ) − = + − f g x x x x 1 1 ; 2 Domain: { } ≠ ≠ x x x 0, 1 ( )( ) ( ) ⋅ = + − f g x x x x 1 1 ; Domain: { } ≠ ≠ x x x 0, 1 ( ) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = + − f g x x x x 1 1 ; Domain: { } ≠ ≠ x x x 0, 1 18. − + − x h 4 1 2 19. (a) Domain: { } − ≤ ≤ x x 4 3 ; Range: { } − ≤ ≤ y y 3 3 (b) ( ) 0, 0 (c) −1 (d) −4 (e) { } < ≤ x x 0 3 (f) y 5 x 5 (21, 23) (1, 21) (3, 0) (6, 3) (g) y 5 x 10 (24, 21) (28, 23) (0, 0) (6, 3) (h) y 5 x 5 (0, 0) (3, 23) (24, 3) (22, 1) (e) No symmetry (f) Neither (g) ( ) ( ) ( ) −3, 0 , 0, 0 , 3, 0 21. Odd 22. Even 23. Neither 24. Odd 25. 20 23 220 3 Local maximum value: 4.04 at = − x 0.91 Local minimum value: −2.04 at = x 0.91 Increasing: 3, 0.91 , 0.91, 3 [ ] [ ] − − Decreasing: [ ] −0.91, 0.91 26. 40 22 26 3 Local maximum value: 1.53 at = x 0.41 Local minimum values: 0.54 at = − x 0.34 and −3.56 at = x 1.80 Increasing: [ ] [ ] −0.34, 0.41 , 1.80, 3 Decreasing: 2, 0.34 , 0.41, 1.80 [ ] [ ] − − 33. y 5 x 5 (24, 4) (4, 4) (0, 0) 27. (a) 23 (b) 7 (c) 47 28. −5 29. −17 30. = − + y x 17 24 31. No 32. Yes 34. y 5 x 5 (4, 2) (1, 1) (0, 0) 35. y 5 x 5 (0, 24) (22, 22) (24, 0) (2, 22) (4, 0) Intercepts: 4,0, 4,0, 0, 4 ( ) ( ) ( ) − − Domain: all real numbers Range: { } [ ) ≥ − − ∞ y y 4 or 4, 36. y 2 x 5 (0, 0) (22, 24) (2, 24) Intercept: ( ) 0, 0 Domain: all real numbers Range: { } ( ] ≤ −∞ y y 0 or , 0 37. y 5 x 7 (1, 0) (5, 2) (2, 1) Intercept: ( ) 1, 0 Domain: { } [ ) ≥ ∞ x x 1 or 1, Range: { } [ ) ≥ ∞ y y 0 or 0, 38. (23, 2) (1, 0) (0, 1) y x 5 5 Intercepts: ( ) 0, 1 , ( ) 1, 0 Domain: { } ( ] ≤ −∞ x x 1 or , 1 Range: { } [ ) ≥ ∞ y y 0 or 0, 39. (1, 2) (2, 3) (0, 3) y x 5 5 Intercept: ( ) 0, 3 Domain: all real numbers Range: { } [ ) ≥ ∞ y y 2 or 2, 40. (0, 224) (23, 26) (23.6, 0) (22, 28) (21, 210) y x 30 3 Intercepts: ( ) − 0, 24, ( ) − −2 4, 0 3 or about ( ) −3.6, 0 Domain: all real numbers Range: all real numbers 41. (a) { } ( ) > − − ∞ x x 2 or 2, (b) ( ) 0, 0 (c) (2, 3) (1, 3) (1, 2) (22, 26) y x 5 5 (d) { } ( ) > − − ∞ y y 6 or 6, 42. (a) { } [ ) ≥ − − ∞ x x 4 or 4, (b) ( ) 0, 1 (c) y x 5 5 (1, 3) (0, 1) (24, 24) (d) { } − ≤ < > y y y 4 0 or 0 or [ ) ( ) − ∪ ∞ 4, 0 0, Chapter Test (page 135) 1. (a) Domain: { } 2, 4, 6, 8 ; range: { } 5, 6, 7, 8 ; function (b) Domain: { } −3, 1, 4 ; range: { } −2, 3, 5, 7 ; not a function (c) Domain: { } ≥ − x x 1 ; range: all real numbers; not a function (d) Domain: all real numbers; range: { } ≥ y y 2 ; function 2. Domain: { } ( ) ≤ − = x x f 4 5 ; 1 3 3. Domain: { } ( ) ≠ − − = x x g 2 ; 1 1 4. Domain: { } ( ) ≠ − ≠ − = x x x h 9, 4 ; 1 1 8 5. (a) Domain: { } − ≤ ≤ x x 5 5 ; range: { } − ≤ ≤ y y 3 3 (b) ( ) 0, 2 , ( ) −2, 0 , and ( ) 2, 0 (c) ( ) = f 1 3 (d) = − x 5 and = x 3 (e) { } [ ) ( ] − ≤ < − < ≤ − − ∪ x x x 5 2 or 2 5or 5, 2 2,5 6. Local maximum values: ( ) − ≈ − f 0.85 0.86; ( ) ≈ f 2.35 15.55; local minimum value: ( ) = − f 0 2; the function is increasing on the intervals [ ] − − 5, 0.85 and [ ] 0, 2.35 and decreasing on the intervals [ ] −0.85, 0 and [ ] 2.35, 5. 43. = A 11 44. (a) ( ) = + A x x x 2 40 2 (b) 42 ft2 (c) 28 ft2 (d) 0 100 7 A is smallest when ≈ x 2.15 ft. 45. (a) ( ) = − A x x x 10 3 (b) The largest area that can be enclosed by the rectangle is approximately 12.17 square units.
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