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Unformatted text preview: Section 1.1 Chapter 1 Prerequisites for Calculus Section 1.1 Exercises 1. &x ! "1 " 1 ! "2 &y ! "1 " 2 ! "3 ■ Section 1.1 Lines (pp. 1–9) 2. &x ! "1 " ("3) ! 2 &y ! "2 " 2 ! "4 Quick Review 1.1 3. &x ! "8 " ("3) ! "5 &y ! 1 " 1 ! 0 1. y ! "2 # 4(3 " 3) ! "2 # 4(0) ! "2 # 0 ! "2 2. 3 3 2x x 4. &x ! 0 " 0 ! 0 &y ! "2 " 4 ! "6 ! 3 " 2(x # 1) ! 3 " 2x " 2 ! "2 ! "1 2"3 5"4 5. (a, c) y 5 "1 1 3. m ! $$ ! $$ ! "1 2 " ("3) 3 " ("1) B 5 4 4. m ! $$ ! $$ 5 x A !4" 1 5. (a) 3(2) " 4 $$ ! 5 6"1 !5 Yes (b) 3(3) " 4("1) ! 5 13 % 5 1 " ("2) 2"1 3 1 (b) m ! $$ ! $$ ! 3 No 6. (a, c) 6. (a) 7 ! "2("1) # 5 7 ! 2 # 5 Yes y 5 (b) 1 ! "2("2) # 5 1 % 9 No 7. d ! #(x $$ $$x$ " )2$# $$($ y2$ "$y$ )2$ 2 1 1 5 x A ! #(0 $$ "$ 1$ )2$ #$(1$$ "$0$ )2 B ! #2$ 8. d = #(x $$ $$x$ " )2$# $$($ y2$ "$y$ )2$ 2 1 1 ! ! ! ! %(1&&"&2)&&#&!"&&&"&1&" 1 $$ 3 2 2 "2 " ("1) 1 " ("2) 7. (a, c) 1 3 y 5 %(" &1&)&&#&!"&$&3$&" 4 2 2 "1 3 (b) m ! $$ ! $$ ! "$$ B %&&&& 16 9 A 1 # $$ 5 % && 25 $$ 9 x 5 3 ! $$ 9. 4x " 3y ! 7 3"3 "1 " 2 0 "3 (b) m ! $$ ! $$ ! 0 "3y ! "4x # 7 4 3 7 3 y ! $$x " $$ 8. (a, c) y 5 10. "2x # 5y ! "3 A 5y ! 2x " 3 2 5 3 5 y ! $$x " $$ 5 B x 1 2 Section 1.1 8. continued 26. The line contains (0, 0) and (5, 2). "3 " 2 "5 (b) m ! $$ ! $$ (undefined) 1"1 0 This line has no slope. 9. (a) x ! 2 (b) y ! 3 2"0 5"0 2 y ! $$x 5 2 5 m ! $$ ! $$ 27. 3x # 4y ! 12 10. (a) x ! "1 4y ! "3x # 12 4 3 (b) y ! $$ 3 4 y ! "$$x # 3 11. (a) x ! 0 (b) y ! "#2$ 3 4 (a) Slope: "$$ 12. (a) x ! "! (b) y-intercept: 3 (b) y ! 0 (c) 13. y ! 1(x " 1) # 1 14. y ! "1[x " ("1)] # 1 y ! "1(x # 1) # 1 15. y ! 2(x " 0) # 3 ["10, 10] by ["10, 10] 16. y ! "2[x " ("4)] # 0 y ! "2(x # 4) # 0 28. x # y ! 2 y ! "x # 2 3"0 3 2"0 2 3 y ! $$(x " 0) # 0 2 3 y ! $$x 2 17. m ! $$ ! $$ (a) Slope: "1 (b) y-intercept: 2 (c) 2y ! 3x 3x " 2y ! 0 1"1 2"1 ["10, 10] by ["10, 10] 0 1 18. m ! $$ ! $$ ! 0 y ! 0(x " 1) # 1 y!1 "2 " 0 "2 " ("2) x y 3 4 y x $$ ! "$$ # 1 4 3 4 y ! "$$x # 4 3 29. $$ # $$ ! 1 "2 0 19. m ! $$ ! $$ (undefined) 4 3 (a) Slope: "$$ Vertical line: x ! "2 "3 "2 " 1 4 2 " ("2) 3 y ! "$$[x " ("2)] # 1 4 3 4 20. m ! $$ ! $$ ! "$$ (b) y-intercept: 4 (c) 4y ! "3(x # 2) # 4 4y ! "3x " 2 ["10, 10] by ["10, 10] 30. y ! 2x # 4 3x # 4y ! "2 (a) Slope: 2 21. y ! 3x " 2 22. y ! "1x # 2 or y ! "x # 2 (b) y-intercept: 4 (c) 1 2 23. y ! "$$x " 3 1 3 24. y ! $$x " 1 ["10, 10] by ["10, 10] 25. The line contains (0, 0) and (10, 25). 25 " 0 10 " 0 5 y ! $$x 2 25 10 5 2 m ! $$ ! $$ ! $$ Section 1.1 31. (a) The desired line has slope "1 and passes through (0, 0): y ! "1(x " 0) # 0 or y ! "x. "1 (b) The desired line has slope $$ ! 1 and passes through "1 (0, 0): y ! 1(x " 0) # 0 or y ! x. 39. (a) y ! 0.680x # 9.013 (b) The slope is 0.68. It represents the approximate average weight gain in pounds per month. (c) 32. (a) The given equation is equivalent to y ! "2x # 4. The desired line has slope "2 and passes through ("2, 2): y ! "2(x # 2) # 2 or y ! "2x " 2. "1 "2 1 2 (b) The desired line has slope $$ ! $$ and passes through [15, 45] by [15, 45] (d) When x ! 30, y ' 0.680(30) # 9.013 ! 29.413. She weighs about 29 pounds. 40. (a) y ! 1,060.4233x " 2,077,548.669 ("2, 2): 1 2 3 (b) The slope is 1,060.4233. It represents the approximate rate of increase in earnings in dollars per year. 1 2 y ! $$(x # 2) # 2 or y ! $$x # 3. 33. (a) The given line is vertical, so we seek a vertical line through ("2, 4): x ! "2. (c) (b) We seek a horizontal line through ("2, 4): y ! 4. 34. (a) The given line is horizontal, so we seek a horizontal ! 1 2 " 1 2 line through "1, $$ : y ! $$. ! " 1 (b) We seek a vertical line through "1, $$ : x ! "1. 2 9"2 7 35. m ! $$ ! $$ 3"1 2 7 7 3 f(x) ! $$(x " 1) # 2 ! $$x " $$ 2 2 2 7 3 Check: f(5) ! $$(5) " $$ ! 16, as expected. 2 2 7 3 7 3 Since f(x) ! $$x " $$, we have m ! $$ and b ! "$$. 2 2 2 2 "4 " ("1) "3 3 4"2 2 2 3 3 f(x) ! "$$(x " 2) # ("1) ! "$$x # 2 2 2 3 Check: f(6) ! "$$(6) # 2 ! "7, as expected. 2 3 3 Since f(x) ! "$$x # 2, we have m ! "$$ and b ! 2. 2 2 36. m ! $$ ! $$ ! "$$ 37. 2 3 y"3 4 " ("2) "$$ ! $$ 2 3 "$$(6) ! y " 3 "4 ! y " 3 (d) When x ! 2000, y ' 1,060.4233(2000) " 2,077,548.669 ' 43,298. In 2000, the construction workers’ average annual compensation will be about $43,298. 41. y ! 1 ' (x " 3) # 4 y!x"3#4 y!x#1 This is the same as the equation obtained in Example 5. x c y When x ! 0, we have $$ ! 1, so y ! d. d x (b) When y ! 0, we have $$ ! 2, so x ! 2c. c y When x ! 0, we have $$ ! 2, so y ! 2d. d 42. (a) When y ! 0, we have $$ ! 1, so x ! c. The x-intercept is 2c and the y-intercept is 2d. 2 k 3 k 43. (a) The given equations are equivalent to y ! "$$x # $$ and y ! "x # 1, respectively, so the slopes are 2 k 2 k "$$ and "1. The lines are parallel when "$$ ! "1, so k ! 2. 2 "1 (b) The lines are perpendicular when "$$ ! $$, so k "1 k ! "2. "1 ! y 38. [1975, 1995] by [20,000, 35,000] 2 " ("2) x " ("8) 2 ! $$ 2(x # 8) ! 4 x#8 !2 x ! "6 68 " 69.5 0.4 " 0 "1.5 0.4 44. (a) m ' $$ ! $$ ! "3.75 degrees/inch 10 " 68 4 " 0.4 "58 3.6 5 " 10 4.7 " 4 "5 0.7 (b) m ' $$ ! $$ ' "16.1 degrees/inch (c) m ' $$ ! $$ ! "7.1 degrees/inch (d) Best insulator: Fiberglass insulation Poorest insulator: Gypsum wallboard The best insulator will have the largest temperature change per inch, because that will allow larger temperature differences on opposite sides of thinner layers. 4 Section 1.1 &p &d 10.94 " 1 100 " 0 9.94 100 y 45. Slope: k ! $$ ! $$ ! $$ 6 ! 0.0994 atmospheres per meter (–1, 4) At 50 meters, the pressure is (2, 3) p ! 0.0994(50) # 1 ! 5.97 atmospheres. 46. (a) d(t) ! 45t (–1, 1) (b) x 6 (2, 0) y [0, 6] by ["50, 300] 6 (c) The slope is 45, which is the speed in miles per hour. (d) Suppose the car has been traveling 45 mph for several hours when it is first observed at point P at time t ! 0. (2, 3) (e) The car starts at time t ! 0 at a point 30 miles past P. (–1, 1) 47. (a) y ! 5632x " 11,080,280 6 (2, 0) (–1, –2) (b) The rate at which the median price is increasing in dollars per year (c) y ! 2732x " 5,362,360 50. (d) The median price is increasing at a rate of about $5632 per year in the Northeast, and about $2732 per year in the Midwest. It is increasing more rapidly in the Northeast. y (c, d) W (a, b) 48. (a) Suppose x(F is the same as x(C. X Z 9 5 x ! $$x # 32 (g, h) !1 " $5$"x ! 32 9 4 5 x x (e, f) Y "$$x ! 32 Suppose that the vertices of the given quadrilateral are x ! "40 (a, b), (c, d), (e, f ), and (g, h). Then the midpoints of the Yes, "40(F is the same as "40(C. consecutive sides are W $$, $$ , X $$, $$ , " ! " a#c b#d c#e d#f 2 2 2 2 e#g f#h g#a h#b Y $$, $$ , and Z $$, $$ . When these four 2 2 2 2 ! (b) ! " ! " points are connected, the slopes of the sides of the resulting figure are: ["90, 90] by ["60, 60] It is related because all three lines pass through the point ("40, "40) where the Fahrenheit and Celsius temperatures are the same. 49. The coordinates of the three missing vertices are (5, 2), ("1, 4) and ("1, "2), as shown below. y 6 (2, 3) (5, 2) (–1, 1) (2, 0) 6 x d#f b#d $$ " $$ 2 2 f"b WX: $$ ! $$ c#e a#c e "a $$ " $$ 2 2 f#h d#f $$ " $$ 2 2 h"d XY: $$ ! $$ e#g c#e g"c $$ " $$ 2 2 f#h h#b $$ " $$ 2 2 f"b ZY: $$ ! $$ e#g g#a e "a $$ " $$ 2 2 h#b b#d $$ " $$ 2 2 h"d WZ: $$ ! $$ g#a a#c g"c $$ " $$ 2 2 Opposite sides have the same slope and are parallel. Section 1.1 4"0 3"0 4 3 51. The radius through (3, 4) has slope $$ ! $$. "1 4/3 3 4 3 4 The tangent line is tangent to this radius, so its slope is $$ ! "$$. We seek the line of slope "$$ that passes through (3, 4). 3 4 3 9 y ! "$$x # $$ # 4 4 4 3 25 y ! "$$x # $$ 4 4 y ! "$$(x " 3) # 4 52. (a) The equation for line L can be written as A B C B "1 "A/B A B B A y ! "$$x # $$, so its slope is "$$. The perpendicular line has slope $$ ! $$ and passes through (a, b), so its equation is B A y ! $$(x " a) # b. B A (b) Substituting $$(x " a) # b for y in the equation for line L gives: (A ) B Ax # B $$(x " a) # b ! C A2x # B2(x " a) # ABb ! AC (A2 # B2)x ! B2a # AC " ABb B2a # AC " ABb A #B x !$ 2 $ 2 Substituting the expression for x in the equation for line L gives: B2a # AC " ABb A #B "A(B2a # AC " ABb) C(A2 # B2) By ! $$$ #$ $ 2 2 A #B A2 # B 2 ! " # By ! C A$ 2 $ 2 "AB2a " A2C # A2Bb # A2C # B2C A #B By ! $$$$ 2 2 A2Bb # B2C " AB2a A #B By ! $$$ 2 2 A2b # BC " ABa A #B y ! $$ 2 2 ! B2a # AC " ABb A2b # BC " ABa A #B A #B " , $$ . The coordinates of Q are $$ 2 2 2 2 (c) Distance ! #(x $$ "$ a$ )2$ #$(y$" $$b$ )2 ! %!&$ &&A&&#$ &B &&&&&"&a"&&#&!$ &&&A&&#$ &B &&&&&"&b"& ! $$$$ " # ! $$$$ " %!&&& &&&&& & A #B A #B ! %&! &&&&&&&&&&" &#&!&&&&&&&&&&&" ! %&!$ &&A&&#$ &B&&&&" &#&!$ &&&A&&#$ &B&&&&" ! # %&&&&&& ! $$$ %&& (A # B ) ! %$ &&&A&&#$ &B&&& 2 B2a # AC " ABb 2 2 B2a # AC " ABb " a(A2 # B2) 2 2 A(C " Bb " Aa) 2 2 BC " ABa " B2b 2 $ $ A2 # B2 B(C " Aa " Bb) 2 2 A2(C " Aa " Bb)2 $$ (A2 # B2)2 2 2 (C " Aa " Bb)2 2 *C " Aa " Bb* ! $$ B 2$ #A $2$# $$ *Aa # Bb " C* ! $$ B 2$ #A $2$# $$ 2 B2(C " Aa " Bb)2 $$ (A2 # B2)2 (A2 # B2)(C " Aa " Bb)2 2 A2b # BC " ABa " b(A2 # B2) 2 2 2 2 2 2 AC " ABb " A2a 2 $ $ A2 # B 2 2 A2b # BC " ABa 2 2 5 6 Section 1.2 ■ Section 1.2 Functions and Graphs (pp. 9–19) Exploration 1 Composing Functions 1. y3 ! g " f, y4 ! f " g 2. Domain of y3: ["2, 2] Range of y3: [0, 2] y1: ["4.7, 4.7] by ["2, 4.2] y2: ["4.7, 4.7] by ["2, 4.2] 5. x2 - 16 Solutions to x2 ! 16: x ! "4, x ! 4 Test x ! "6 ("6)2 ! 36 , 16 x2 - 16 is false when x - "4 Test x ! 0: 02 ! 0 - 16 x2 - 16 is true when "4 - x - 4 Test x ! 6: 62 ! 36 , 16 x2 - 16 is false when x , 4. Solution set: ("4, 4) 6. 9 " x2 * 0 Solutions to 9 " x2 ! 0: x ! "3, x ! 3 Test x ! "4: 9 " ("4)2 ! 9 " 16 ! "7 - 0 9 " x2 * 0 is false when x - "3. Test x ! 0: 9 " 02 ! 9 , 0 9 " x2 * 0 is true when "3 - x - 3. Test x ! 4: 9 " 42 ! 9 " 16 ! "7 - 0 9 " x2 * 0 is false when x , 3. Solution set: ["3, 3] 7. Translate the graph of f 2 units left and 3 units downward. 8. Translate the graph of f 5 units right and 2 units upward. y3: 9. (a) ["4.7, 4.7] by ["2, 4.2] 3. Domain of y4: [0, )); Range of y4: ("), 4] y4: (b) f(x) x2 " 5 x2 " 9 (x # 3)(x " 3) x !4 !4 !0 !0 ! "3 or x ! 3 f(x) ! "6 x2 " 5 ! "6 x2 ! "1 No real solution 10. (a) f(x) ! "5 1 $$ ! "5 x 1 x ! "$$ 5 ["2, 6] by ["2, 6] 4. y3 ! y2 ( y1(x)) ! #$ y1(x $)$ ! #4$$ "$x 2$ 2 y4 ! y1( y2(x)) ! 4 " (y2(x)) ! 4 " (#x$)2 ! 4 " x, x * 0 Quick Review 1.2 1. 3x " 1 + 5x # 3 "2x + 4 x * "2 Solution: ["2, )) 2. x(x " 2) , 0 Solutions to x(x " 2) ! 0: x ! 0, x ! 2 Test x ! "1: "1("1 " 2) ! 3 , 0 x(x " 2) , 0 is true when x - 0. Test x ! 1: 1(1 " 2) ! "1 - 0 x(x " 2) , 0 is false when 0 - x - 2. Test x ! 3: 3(3 " 2) ! 3 , 0 x(x " 2) , 0 is true when x , 2. Solution set: ("), 0) ! (2, )) 3. *x " 3* + 4 "4 + x " 3 + 4 "1 + x + 7 Solution set: ["1, 7] 4. *x " 2* * 5 x " 2 + "5 or x " 2 * 5 x + "3 or x * 7 Solution set: ("), "3] ! [7, )) (b) f(x) = 0 1 $$ ! 0 x No solution 11. (a) (b) 12. (a) (b) f(x) = 4 #x$$ #$7 ! 4 x # 7 ! 16 x !9 Check: #9$$ #$7 ! #1$6$ ! 4; it checks. f(x) ! 1 #x$$ #$7 ! 1 x#7 !1 x ! "6 Check: #" $6$$ #$7 ! 1; it checks. f(x) #x$$ "$1 x"1 x 3 3 ! "2 ! "2 ! "8 ! "7 f(x) ! 3 #x$" $$1 ! 3 x " 1 ! 27 x ! 28 Section 1.2 Section 1.2 Exercises d 2 !2" 7 8. (a) Since we require "x * 0, the domain is ("), 0]. !d 4 2 1. Since A ! !r2 ! ! $$ , the formula is A ! $$, where A (b) ("), 0] (c) represents area and d represents diameter. 2. Let h represent height and let s represent side length. s 2 !2" h2 # $$ ! s2 ["10, 3] by ["4, 2] (d) None 1 4 h2 ! s2 " $$s2 3 4 9. (a) Since we require 3 " x * 0, the domain is ("), 3]. h2 ! $$s2 (b) [0, )) #3$ (c) h ! .$$s 2 Since side length and height must be positive, the formula #$3 is h ! $$s. 2 ["4.7, 4.7] by ["6, 6] (d) None s s h s 2 s 2 10. (a) Since we require x " 2 % 0, the domain is ("), 2) ! (2, )). 1 x"2 (b) Since $$ can assume any value except 0, the range is ("), 0) ! (0, )). s 3. S ! 6e2, where S represents surface area and e represents edge length. (c) 4 4. V ! $$!r3, where V represents volume and r represents 3 radius. 5. (a) ("), )) or all real numbers ["4.7, 4.7] by ["6, 6] (b) ("), 4] (d) None (c) 11. (a) ("), )) or all real numbers (b) ("), )) or all real numbers (c) ["5, 5] by ["10, 10] (d) Symmetric about y-axis (even) 6. (a) ("), )) or all real numbers ["6, 6] by ["3, 3] (b) ["9, )) (d) None (c) 12. (a) ("), )) or all real numbers (b) The maximum function value is attained at the point (0, 1), so the range is ("), 1]. ["5, 5] by ["10, 10] (c) (d) Symmetric about the y-axis (even) 7. (a) Since we require x " 1 * 0, the domain is [1, )). (b) [2, )) ["6, 6] by ["3, 3] (c) (d) Symmetric about the y-axis (even) ["3, 10] by ["3, 10] (d) None 8 Section 1.2 x3, so its domain is 18. (a) This function is equivalent to y ! #$ [0, )). 13. (a) Since we require "x * 0, the domain is ("), 0]. (b) [0, )) (b) [0, )) (c) (c) ["10, 3] by ["1, 2] ["2, 5] by ["2, 8] (d) None (d) None 14. (a) Since we require x % 0, the domain is ("), 0) ! (0, )). 1 x 19. Even, since the function is an even power of x. 1 x (b) Note that $$ can assume any value except 0, so 1 # $$ 20. Neither, since the function is a sum of even and odd powers of x. 21. Neither, since the function is a sum of even and odd powers of x (x1 # 2x0). can assume any value except 1. The range is ("), 1) ! (1, )). 22. Even, since the function is a sum of even powers of x (x2 " 3x0). (c) 23. Even, since the function involves only even powers of x. 24. Odd, since the function is a sum of odd powers of x. 25. Odd, since the function is a quotient of an odd function (x3) and an even function (x2 " 1). ["4, 4] by ["4, 4] (d) None 15. (a) Since we require 4 " x2 * 0, the domain is ["2, 2]. (b) Since 4 " x2 will be between 0 and 4, inclusive (for x in the domain), its square root is between 0 and 2, inclusive. The range is [0, 2]. (c) 26. Neither, since, (for example), y("2) ! 41/3 and y(2) ! 0. 27. Neither, since, (for example), y("1) is defined and y(1) is undefined. 28. Even, since the function involves only even powers of x. 29. (a) ["9.4, 9.4] by ["6.2, 6.2] ["4.7, 4.7] by ["3.1, 3.1] (d) Symmetric about the y-axis (even) Note that f(x) ! "*x " 3* # 2, so its graph is the graph of the absolute value function reflected across the x-axis and then shifted 3 units right and 2 units upward. 3 x2, so its domain is 16. (a) This function is equivalent to y ! #$ all real numbers. (b) [0, )) (b) ("), )) (c) ("), 2] (c) 30. (a) The graph of f(x) is the graph of the absolute value function stretched vertically by a factor of 2 and then shifted 4 units to the left and 3 units downward. ["2, 2] by ["1, 2] (d) Symmetric about the y-axis (even) 17. (a) Since we require x2 % 0, the domain is ("), 0) ! (0, )) 1 x (b) Since $$2 , 0 for all x, the range is (1, )). ["10, 5] by ["5, 10] (b) ("), )) or all real numbers (c) ["3, )) 31. (a) (c) ["4, 4] by ["1, 5] (d) Symmetric about the y-axis (even) ["4.7, 4.7] by ["1, 6] (b) ("), )) or all real numbers (c) [2, )) Section 1.2 32. (a) 44. Line through ("1, 0) and (0, "3): "3 1 "3 " 0 0 " ("1) m ! $$ ! $$ ! "3, so y ! "3x " 3 Line through (0, 3) and (2, "1): "1 " 3 2"0 "4 2 m ! $$ ! $$ ! "2, so y ! "2x # 3 ["4, 4] by ["2, 3] (b) ("), )) or all real numbers f(x) ! (c) [0, )) 33. (a) + "3x " 3, "2x # 3, "1 - x + 0 0-x+2 45. Line through ("1, 1) and (0, 0): y ! "x Line through (0, 1) and (1, 1): y ! 1 Line through (1, 1) and (3, 0): ["3.7, 5.7] by ["4, 9] 0"1 "1 1 3"1 2 2 1 1 3 so y ! "$$(x " 1) # 1 ! "$$x # $$ 2 2 2 m ! $$ ! $$ ! "$$, (b) ("), )) or all real numbers (c) ("), )) or all real numbers + 34. (a) "x, "1 + x - 0 0-x+1 f(x) ! 1, 1 3 "$$x # $$, 2 2 ["2.35, 2.35] by ["1, 3] 1 2 46. Line through ("2, "1) and (0, 0): y ! $$x (b) ("), )) or all real numbers Line through (0, 2) and (1, 0): y ! "2x # 2 (c) [0, )) 35. Because if the vertical line test holds, then for each x-coordinate, there is at most one y-coordinate giving a point on the curve. This y-coordinate would correspond to the value assigned to the x-coordinate. Since there is only one y-coordinate, the assignment would be unique. 36. If the curve is not y ! 0, there must be a point (x, y) on the curve where y % 0. That would mean that (x, y) and (x, "y) are two different points on the curve and it is not the graph of a function, since it fails the vertical line test. 37. No Line through (1, "1) and (3, "1): y ! "1 + 1 $$x, 2 40. No !2 " T 47. Line through $$, 0 and (T, 1): 1"0 T " (T/2) x, "x # 2, + 0+x+1 1-x+2 0"1 5"2 1 3 5 3 so y ! "$$(x " 2) # 1 ! "$$x # $$ + 1 3 5 3 "$$x # $$, 2 T ! T 2 " 2 T A, T 2 0 + x + $$ T $$ - x + T 2 T 2 0 + x - $$ T $$ + x - T 2 3T T + x - $$ 2 3T $$ + x + 2T 2 49. (a) f (g(x)) ! (x 2 " 3) # 5 ! x 2 # 2 (b) g( f (x))! (x # 5)2 " 3 ! (x 2 # 10x # 25) " 3 ! x 2 # 10x # 22 "1 3 1 3 Line through (2, 1) and (5, 0): m ! $$ ! $$ ! "$$, "x # 2, + A, "A, 43. Line through (0, 2) and (2, 0): y ! "x # 2 1 3 T "A, 48. f (x) ! 0+x-1 1+x-2 2+x-3 3+x+4 2, 0, 42. f(x) ! 2, 0, 2 T m ! $$ ! $$, so y ! $$ x " $$ # 0 ! $$x " 1 41. Line through (0, 0) and (1, 1): y ! x Line through (1, 1) and (2, 0): y ! "x # 2 + 0-x+1 1-x+3 "1, 0, f (x) ! 2 $$x " 1, 39. Yes f(x) ! "2 + x + 0 f(x) ! "2x # 2, + 38. Yes f (x) ! 1-x-3 0-x+2 2-x+5 (c) f (g(0)) ! 02 # 2 ! 2 (d) g( f (0)) ! 02 # 10 ' 0 # 22 ! 22 (e) g(g("2)) ! [("2)2 " 3]2 " 3 ! 12 " 3 ! "2 (f) f ( f(x)) ! (x # 5) # 5 ! x # 10 9 10 Section 1.2 50. (a) f (g(x)) ! (x " 1) # 1 ! x g " f: (b) g( f (x)) ! (x # 1) " 1 ! x (c) f (g(x)) ! 0 (d) g( f (0)) ! 0 (e) g(g("2)) ! ("2 " 1) " 1 ! "3 " 1 ! "4 (f) f ( f (x)) ! (x # 1) # 1 ! x # 2 ["4.7, 4.7] by ["2, 4] 51. (a) Enter y1 ! f(x) ! x " 7, y2 ! g(x) ! #x$, y3 ! ( f " g)(x) ! y1(y2(x)), and y4 ! (g " f )(x) ! y2(y1(x)) f " g: g " f: Domain: ("), "1] ! [1, )) Range: [0, )) $$2)2 " 3 (b) ( f " g)(x) ! (#x$# ! (x # 2) " 3, x * "2 ! x " 1, x * "2 (g " f )(x) ! #$ (x2$$ "$3$ )$ #$ 2 ! #$ x2$ "$1 2x " 1 x#3 3x # 1 2"x 54. (a) Enter y1(x) ! f(x) ! $$, y2 ! $$, ["10, 70] by ["10, 3] Domain: [0, )) Range: ["7, )) (b) ( f " g)(x) ! #x$ " 7 ["3, 20] by ["4, 4] Domain: [7, )) Range: [0, )) y3 ! ( f " g)(x) ! y1(y2(x)), and y4 ! (g " f )(x) ! y2(y1(x)). Use a “decimal window” such as the one shown. f " g: (g " f )(x) ! #x$" $$7 52. (a) Enter y1 ! f(x) ! 1 " x 2, y2 ! g(x) ! #x$, y3 ! ( f " g)(x) ! y1(y2(x)), and y4 ! (g " f )(x) ! y2(y1(x)) f " g: ["9.4, 9.4] by ["6.2, 6.2] Domain: ("), 2) ! (2, )) Range: ("), 2) ! (2, )) g " f: ["6, 6] by ["4, 4] Domain: [0, )) Range: ("), 1] g " f: ["9.4, 9.4] by ["6.2, 6.2] Domain: ("), "3) ! ("3, )) Range: ("), "3) ! ("3, )) ["2.35, 2.35] by ["1, 2.1] Domain: ["1, 1] Range: [0, 1] (b) ( f " g)(x) ! 1 " (#x$)2 ! 1 " x, x * 0 (g " f )(x) ! #$1$ "$x2$ 53. (a) Enter y1 ! f(x) ! x 2 " 3, y2 ! g(x) ! #x$# $$2, y3 ! ( f " g)(x) ! y1(y2(x)), and y4 ! (g " f)(x) ! y2(y1(x)). f " g: ! 3x # 1 " 2 $$ " 1 2"x (b) ( f " g)(x) ! $$ 3x # 1 $$ # 3 2"x 2(3x # 1) " (2 " x) (3x # 1) # 3(2 " x) ! $$$, x % 2 7x 7 ! $$, x % 2 ! x, x % 2 ! 2x " 1 " 3 $$ # 1 x#3 (g " f )(x) ! $$ 2x " 1 2 " $$ x#3 3(2x " 1) # (x # 3) ! $$$, x % "3 2(x # 3) " (2x " 1) 7x 7 ["10, 10] by ["10, 10] ! $$, x % "3 Domain: ["2, )) Range: ["3, )) ! x, x % "3 Section 1.2 55. 11 58. [!5, 5] by [!2, 5] [!2.35, 2.35] by [!1.55, 1.55] We require x 2 ! 4 " 0 (so that the square root is defined) We require x 2 ! 1 # 0, so the domain is and x 2 ! 4 # 0 (to avoid division by zero), so the domain (!$, !1) ! (!1, 1) ! (1, $). For values of x in the is (!$, !2) ! (2, $). For values of x in the domain, x2 ! 4 domain, x 2 ! 1 can attain any value in [!1, 0) ! (0, $), so %$ can attain any positive !and hence "#x2#!#4 and % !#4 "# x# "# x 2# !#1 can also attain any value in [!1, 0) ! (0, $). value, so the range is (0, $). (Note that grapher failure may cause the range to appear as a finite interval on a Therefore, % 3 % can attain any value in "x#! ##1 (!$, !1] ! (0, $). The range is (!$, !1] ! (0, $). grapher. (Note that grapher failure can cause the intervals in the 1 2 56. 3 1 range to appear as finite intervals on a grapher.) 59. (a) y 1.5 [!5, 5] by [!2, 5] We require 9 ! x 2 " 0 (so that the fourth root is defined) –2 0 2 x and 9 ! x 2 # 0 (to avoid division by zero), so the d...
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