Appendix 1: Homework Answers for Chapter 1

Section 1.1 Answers

  1. The required points \;A(-3, -7), \;B(1.3, -2), \;C(\pi, \sqrt{10}), \;D(0, 8), \;E(-5.5, 0), \;F(-8, 4), \;G(9.2, -7.8), and H(7, 5) are plotted in the Cartesian Coordinate Plane below.
    The cartesian coordinate plane with points C and H plotted in the first quadrant. Point D is on the positive y-axis. Point F is in the second quadrant. Point E is on the negative x-axis. Point A is in the third quadrant and points B and G are in the fourth quadrant.
  2. Points
    1. The point A(-3, -7) is
      • in Quadrant III
      • symmetric about the x-axis with (-3,7)
      • symmetric about y-axis with (3, -7)
      • symmetric about origin with (3, 7)
    2. The point B(1.3, -2) is
      • in Quadrant IV
      • symmetric about x-axis with (1.3, 2)
      • symmetric about y-axis with (-1.3, -2)
      • symmetric about origin with (-1.3, 2)
    3. The point C(\pi, \sqrt{10}) is
      • in Quadrant I
      • symmetric about x-axis with (\pi, -\sqrt{10})
      • symmetric about y-axis with (-\pi, \sqrt{10})
      • symmetric about origin with (-\pi, -\sqrt{10})
    4. The point D(0, 8) is
      • on the positive y-axis
      •  symmetric about x-axis with (0, -8)
      • symmetric about y-axis with (0, 8)
      • symmetric about origin with (0, -8)
    5. The point E(-5.5, 0) is
      • on the negative x-axis
      • symmetric about x-axis with (-5.5, 0)
      • symmetric about y-axis with (5.5, 0)
      • symmetric about origin with (5.5, 0)
    6. The point F(-8, 4) is
      • in Quadrant II
      • symmetric about x-axis with (-8, -4)
      • symmetric about y-axis with (8, 4)
      • symmetric about origin with (8, -4)
    7. The point G(9.2, -7.8) is
      • in Quadrant IV
      • symmetric about x-axis with (9.2, 7.8)
      • symmetric about y-axis with (-9.2, -7.8)
      • symmetric about origin with (-9.2, 7.8)
    8. The point H(7, 5) is
      • in Quadrant I
      • symmetric about x-axis with (7, -5)
      • symmetric about y-axis with (-7, 5)
      • symmetric about origin with (-7, -5)
  3. d = 5 units, M = \left(-1, \frac{7}{2} \right)
  4. d = 4 \sqrt{10} units, M = \left(1, -4 \right)
  5. d = \sqrt{26} units, M = \left(1, \frac{3}{2} \right)
  6. d= \frac{\sqrt{37}}{2} units, M = \left(\frac{5}{6}, \frac{7}{4} \right)
  7. d = \sqrt{74} units, M = \left(\frac{13}{10}, -\frac{13}{10} \right)
  8. d= 3\sqrt{5} units, M = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{3}}{2} \right)
  9. d = \sqrt{83} units, M = \left(4 \sqrt{5}, \frac{5 \sqrt{3}}{2} \right)
  10. d = 2 units, M = \left( 0, 0\right)
  11. (-3, -4), 5 miles, (4, -4)
  12. The distance from A to B is |AB| = \sqrt{13}, the distance from A to C is |AC| = \sqrt{52}, and the distance from B to C is |BC| = \sqrt{65}. because \left(\sqrt{13}\right)^2 + \left( \sqrt{52} \right)^2 = \left( \sqrt{65} \right)^2, we are guaranteed by the converse of the Pythagorean Theorem that the triangle is a right triangle.

Section 1.2 Answers

  1. The mapping M is not a function because `Tennant’ is matched with both `Eleven’ and `Twelve.’
  2. The mapping C is a function because each input is matched with only one output. The domain of C is \{ \text{Hartnell}, \text{Cushing}, \text{Hurndall}, \text{Troughton} \} and the range is \{\text{One}, \text{Two} \}. We can represent C as the following set of ordered pairs: \{ (\text{Hartnell}, \text{One}), (\text{Cushing}, \text{One}), (\text{Hurndall}, \text{One}), (\text{Troughton}, \text{Two}) \}
  3. In this case, y is a function of x because each x is matched with only one y.The domain is \{ -3, -2, -1,0,1,2,3 \} and the range is \{ 0,1,2,3 \}.As ordered pairs, this function is \{ (-3,3), (-2,2), (-1,1), (0,0), (1,1), (2,2), (3,3) \}
  4. In this case, y is not a function of x because there are x values matched with more than one y value. For instance, 1 is matched both to 1 and -1.
  5. The mapping is a function because given any word, there is only one answer to `how many letters are in the word?’ For the range, we would need to know what the length of the longest word is and whether or not we could find words of all the lengths between 1 (the length of the word `a’) and it. See here.
  6. because Grover Cleveland was both the 22nd and 24th POTUS, neither mapping described in this exercise is a function.
  7. The outdoor temperature could never be the same for more than two different times – so, for example, it could always be getting warmer or it could always be getting colder.
  8. f(2) = \frac{7}{4}, f(x) = \frac{2x+3}{4}
  9. f(2) = \frac{5}{2}, f(x) = \frac{2(x+3)}{4} = \frac{x+3}{2}
  10. f(2) = 7, f(x) = 2\left(\frac{x}{4} + 3\right) = \frac{1}{2} x + 6
  11. f(2) = \sqrt{7}, f(x) = \sqrt{2x+3}
  12. f(2) = \sqrt{10}, f(x) = \sqrt{2(x+3)} = \sqrt{2x+6}
  13. f(2) = 2 \sqrt{5}, f(x) = 2\sqrt{x+3}
  14. For f(x) = 2x+1
    • f(3) = 7
    • f(-1) = -1
    • f\left(\frac{3}{2} \right) = 4
    • f(4x) = 8x+1
    • 4f(x) = 8x+4
    • f(-x) = -2x+1
    • f(x-4) = 2x-7
    • f(x) - 4 = 2x-3
    • f\left(x^2\right) = 2x^2+1
  15. For f(x) = 3-4x
    • f(3) = -9
    • f(-1) = 7
    • f\left(\frac{3}{2} \right) = -3
    • f(4x) = 3-16x
    • 4f(x) = 12-16x
    • f(-x) = 4x+3
    • f(x-4) = 19-4x
    • f(x) - 4 = -4x-1
    • f\left(x^2\right) = 3-4x^2
  16. For f(x) = 2 - x^2
    • f(3) = -7
    • f(-1) = 1
    • f\left(\frac{3}{2} \right) = -\frac{1}{4}
    • f(4x) = 2-16x^2
    • 4f(x) = 8-4x^2
    • f(-x) = 2-x^2
    • f(x-4) = -x^2+8x-14
    • f(x) - 4 = -x^{2} - 2
    • f\left(x^2\right) = 2-x^4
  17. For f(x) = x^2 - 3x + 2
    • f(3) = 2
    • f(-1) = 6
    • f\left(\frac{3}{2} \right) = -\frac{1}{4}
    • f(4x) = 16x^2-12x+2
    • 4f(x) = 4x^2-12x+8
    • f(-x) = x^2+3x+2
    • f(x-4) = x^2-11x+30
    • f(x) - 4 = x^2-3x-2
    • f\left(x^2\right) = x^4-3x^2+2
  18. For f(x) = 6
    • f(3) = 6
    • f(-1) =6
    • f\left(\frac{3}{2} \right) = 6
    • f(4x) = 6
    • 4f(x) = 24
    • f(-x) = 6
    • f(x-4) = 6
    • f(x) - 4 = 2
    • f\left(x^2\right) = 6
  19. For f(x) = 0
    • f(3) = 0
    • f(-1) =0
    • f\left(\frac{3}{2} \right) = 0
    • f(4x) = 0
    • 4f(x) = 0
    • f(-x) = 0
    • f(x-4) = 0
    • f(x) - 4 = -4
    • f\left(x^2\right) = 0
  20. For f(x) = 2x-5
    • f(2) = -1
    • f(-2) = -9
    • f(2a) = 4a-5
    • 2 f(a) = 4a-10
    • f(a+2) = 2a-1
    • f(a) + f(2) = 2a-6
    • f \left( \frac{2}{a} \right) = \frac{4}{a} - 5 = \frac{4-5a}{a}
    • \frac{f(a)}{2} =\frac{2a-5}{2}
    • f(a + h) = 2a + 2h - 5
  21. For f(x) = 5-2x
    • f(2) = 1
    • f(-2) = 9
    • f(2a) = 5-4a
    • 2 f(a) = 10-4a
    • f(a+2) = 1-2a
    • f(a) + f(2) = 6-2a
    • f \left( \frac{2}{a} \right) = 5 - \frac{4}{a}= \frac{5a-4}{a}
    • \frac{f(a)}{2} = \frac{5-2a}{2}
    • f(a + h) = 5-2a-2h
  22. For f(x) = 2x^2-1
    • f(2) = 7
    • f(-2) = 7
    • f(2a) = 8a^2-1
    • 2 f(a) = 4a^2-2
    • f(a+2) = 2a^2+8a+7
    • f(a) + f(2) = 2a^2+6
    • f \left( \frac{2}{a} \right) = \frac{8}{a^2} - 1= \frac{8-a^2}{a^2}
    • \frac{f(a)}{2} = \frac{2a^2-1}{2}
    • f(a + h) = 2a^2+4ah+2h^2-1
  23. For f(x) = 3x^2+3x-2
    • f(2) = 16
    • f(-2) = 4
    • f(2a) = 12a^2+6a-2
    • 2 f(a) = 6a^2+6a-4
    • f(a+2) = 3a^2+15a+16
    • f(a) + f(2) = 3a^2+3a+14
    • f \left( \frac{2}{a} \right) = \frac{12}{a^2} + \frac{6}{a} - 2 = \frac{12+6a-2a^2}{a^2}
    • \frac{f(a)}{2} = \frac{3a^2+3a-2}{2}
    • f(a + h) = 3a^2 + 6ah + 3h^2+3a+3h-2
  24. For f(x) = 117
    • f(2) = 117
    • f(-2) = 117
    • f(2a) = 117
    • 2 f(a) = 234
    • f(a+2) = 117
    • f(a) + f(2) = 234
    • f \left( \frac{2}{a} \right) = 117
    • \frac{f(a)}{2} = \frac{117}{2}
    • f(a + h) = 117
  25. For f(x) = \frac{x}{2}
    • f(2) = 1
    • f(-2) = -1
    • f(2a) = a
    • 2 f(a) = a
    • f(a+2) = \frac{a+2}{2}
    • f(a) + f(2) = \frac{a}{2}+ 1 = \frac{a+2}{2}
    • f \left( \frac{2}{a} \right) = \frac{1}{a}
    • \frac{f(a)}{2} = \frac{a}{4}
    • f(a + h) = \frac{a+h}{2}
  26. For f(x) = 2x-1, f(0) = -1 and f(x) = 0 when x = \frac{1}{2}
  27. For f(x) = 3 - \frac{2}{5} x, f(0) = 3 and f(x) = 0 when x = \frac{15}{2}
  28. For f(x) = 2x^2-6, f(0) = -6 and f(x) = 0 when x = \pm \sqrt{3}
  29. For f(x) = x^2-x-12, f(0) = -12 and f(x) = 0 when x = -3 or x=4
  30. Function
  31. Function
  32. Function
  33. Not a function
  34. Function
  35. Not a function
  36. Not a function
  37. Function
  38. Not a function
  39. Function
  40. Not a function
  41. Function
  42. Function
  43. Function
  44. Not a function
  45. Function, domain = \{-3, -2, -1, 0, 1, 2 ,3\}, range = \{0, 1, 4, 9 \}
  46. Not a function
  47. Function, domain = \left\{ -7, -3, 3, 4, 5, 6 \right\}, range = \left\{ 0,4,5,6,9 \right\}
  48. Function, domain = \left\{ 1, 4, 9, 16, 25, 36, \ldots \right\} \\ = \left\{ x \, | \, x \text{ is a perfect square} \right\}, range = \left\{ 2, 4, 6, 8, 10, 12, \ldots \right\} \\ = \left\{ y \, | \, y \text{ is a positive even integer} \right\}
  49. Not a function
  50. Function, domain = \{x \, | \, x \text{ is irrational} \}, range = \{ 1\}
  51. Function, domain = \{x \, | \, 1, 2, 4, 8, \ldots \} = \{x \, | \, x=2^{n} \text{ for some whole number } n \}, range = \{ 0, 1, 2, 3, \ldots \} = \{y \, | \, y \text{ is any whole number}\}
  52. Function, domain = \{x \, | \, x \text{ is any integer} \}, range = \{y \, | \, y \text{ is the square of an integer}\}
  53. Not a function
  54. Function, domain = \{x \, | \, -2 \leq x < 4 \} = [-2, 4),  range = \{3\}
  55. Function, domain = \{x \, | \, x \text{ is a real number} \} = (-\infty, \infty), range = \{y \, | \, y \geq 0 \} = [0,\infty)
  56. Not a function
  57. Horizontal Line Test: A graph on the xy-plane represents x as a function of y if and only if no horizontal line intersects the graph more than once.
  58. Function, domain = \{-4, -3, -2, -1, 0, 1\},  range = \{-1, 0, 1, 2, 3, 4\}
  59. Not a function
  60. Function, domain = (-\infty, \infty), range = [1, \infty)
  61. Not a function
  62. .
    • Number 58 represents x as a function of y, domain = \{-1, 0, 1, 2, 3, 4\} and range = \{-4, -3, -2, -1, 0, 1 \}
    • Number 61 represents x as a function of y, domain = (-\infty, \infty) and range = [1, \infty)
  63. Function, domain = [2, \infty), range = [0, \infty)
  64. Function, domain = (-\infty, \infty),  range = (0, 4]
  65. Not a function
  66. Function, domain = [-5,-3) \cup(-3, 3), range = (-2, -1) \cup [0, 4)
  67. Only number 63 represents v as a function of w; domain = [0, \infty) and range = [2, \infty)
  68. Function, domain = [-2, \infty), range = [-3, \infty)
  69. Not a function
  70. Function, domain = (-5, 4), range = (-4, 4)
  71. Function , domain = [0,3) \cup (3,6], range = (-4,-1] \cup [0,4]
  72. None of numbers 68 – 71 represent t as a function of T.
  73. Function, domain = (-\infty, \infty), range = (-\infty, 4]
  74. Function, domain = (-\infty, \infty), range = (-\infty, 4]
  75. Function, domain = [-2, \infty), range = (-\infty, 3]
  76. Function, domain = (-\infty, \infty), range = (-\infty, \infty)
  77. Only number 75 represents s as a function of H; domain = (-\infty, 3] and range = [-2, \infty)
  78. Function, domain = (-\infty, 0] \cup (1, \infty), range = (-\infty, 1] \cup \{ 2\}
  79. Function,  domain = [-3,3], range = [-2,2]
  80. Not a function
  81. Function, domain = (-\infty, \infty), range = \{2\}
  82. Only number 80 represents t as a function of u; domain = (-\infty, \infty) and range=\{2 \}
  83. f(-2) = 2
  84. g(-2) = -5
  85. f(2) = 3
  86. g(2) = 3
  87. f(0) = -1
  88. g(0) = 0
  89. x = -4, -1, 1
  90. t = -4, 0, 4
  91. Domain: [-5,3], Range: [-5,4]
  92. Domain: [-4,4], Range: [-5,5)
  93. f(x) =2-x, Domain: (-\infty, \infty), Range: (-\infty, \infty)
    The line f(x) on the cartesian plane. The line decreases left to right and has a horizontal intercept at x = 2 and a vertical intercept at y = 2.
  94. g(t) = \dfrac{t - 2}{3}, Domain: (-\infty, \infty), Range: (-\infty, \infty)
    The line g(t) on the cartesian plane. The line increases left to right and intersects the vertical axis at y = -2/3 and intersects the horizontal axis at x=2.
  95. h(s) = s^2+1, Domain: (-\infty, \infty), Range: [1, \infty)
    The curve h(s) on the cartesian plane. The graph is a parabola that opens up. The vertex of the parabola is at (0,1).
  96. f(x) = 4-x^2, Domain: (-\infty, \infty), Range: (-\infty, 4]
    The curve f(x) on the cartesian plane. The curve is a parabola that opens downward. The vertex of the parabola is at (0,4).
  97. g(t) = 2, Domain: (-\infty, \infty), Range: \{2\}
    The line g(t) on the cartesian plane. The line is a horizontal line that intersects the vertical axis at y = 2.
  98. h(s) = s^3, Domain: (-\infty, \infty), Range: (-\infty, \infty)
    The curve of h(s) on the cartesian plane. The curve increases from negative infinity to positive infinity and intersects both axes at the origin.
  99. f(x) = x(x-1)(x+2), Domain: (-\infty, \infty), Range: (-\infty, \infty)
    The curve f(x) on the cartesian plane. The curve intersects the horizontal axis at x = -2, 0, and 1. The graph increases until around x = -1, then decreases until around x = 0.5 and then changes back to increasing.
  100. g(t) = \sqrt{t-2}, Domain: [2, \infty), Domain: [0, \infty)
    The curve g(t) on the cartesian plane. The graph starts at the point (2,0) and increases to the right .
  101. h(s) = \sqrt{5 - s}, Domain: (-\infty, 5], Range: [0, \infty)
    The curve h(s) on the cartesian plane. The graph decreases from positive infinity to the point (5,0).
  102. f(x) = 3-2\sqrt{x+2}, Domain: [-2,\infty), Range: (-\infty, 3]
    The curve f(x) on the cartesian plane. The curve starts at (-2,3) and decreases to negative infinity.
  103. g(t) = \sqrt[3]{t}, Domain: (-\infty, \infty), Range: (-\infty, \infty)
    The curve g(t) on the cartesian plane. The curve increases from left to right, at a very slow rate. The curve intersects both axes at (0,0).
  104. h(s) = \dfrac{1}{s^{2} + 1}, Domain: (-\infty, \infty), Range: (0, 1]
    The curve h(s) on the cartesian plane. The curve remains above the horizontal axis at all times. The graph increases to the point (0,1) and then decreases back towards the horizontal axis.
  105. .
    1. domain = \{ -1, 0, 1, 2 \}, range = \{ -3, 0, 4\}
    2. f(0) = -3, f(x) = 0 for x = -1, 1
    3. f = \{ (-1,0), (0, -3), (1,0), (2,4) \}
    4. The points (-1,0), (0,-3), (1,0) and (2,4) plotted on the cartesian plane.
  106. .
    1. domain = \{ -1, 0, 2, 3 \}, range =\{ 2, 3, 4 \}
    2. The mapping g. There is a line from -1 in the domain to 4 in the range, 0 from the domain to 2 in the range, from 2 in the domain to 3 in the range, and from 3 in the domain to 4 in the range.
    3. Find g(0) = 2 and g(x) = 0 has no solutions.
    4. The points (-1,4), (0,2), (2,3), and (3,4) plotted on the cartesian plane.
  107. F(4) = 4^2 = 16 (when t = 4), the solutions to F(x) = 4 are x = \pm 2 (when t = \pm 2).
  108. G(4) = 7 (when t = 2), the solution to G(t) = 4 is x = -2 (when t = -1)
  109. A(3) = 9, so the area enclosed by a square with a side of length 3 inches is 9 square inches. The solutions to A(\ell) = 36 are \ell = \pm 6. because \ell is restricted to \ell > 0, we only keep \ell = 6. This means for the area enclosed by the square to be 36 square inches, the length of the side needs to be 6 inches. because \ell represents a length, \ell > 0.
  110. A(2) = 4\pi, so the area enclosed by a circle with radius 2 meters is 4\pi square meters. The solutions to A(r) = 16\pi are r = \pm 4. because r is restricted to r > 0, we only keep r = 4. This means for the area enclosed by the circle to be 16\pi square meters, the radius needs to be 4 meters. because r represents a radius (length), r > 0.
  111. V(5) = 125, so the volume enclosed by a cube with a side of length 5 centimeters is 125 cubic centimeters. The solution to V(s) = 27 is s = 3. This means for the volume enclosed by the cube to be 27 cubic centimeters, the length of the side needs to 3 centimeters. because x represents a length, x > 0.
  112. V(3) = 36\pi, so the volume enclosed by a sphere with radius 3 feet is 36\pi cubic feet. The solution to V(r) = \frac{32\pi}{3} is r = 2. This means for the volume enclosed by the sphere to be \frac{32\pi}{3} cubic feet, the radius needs to 2 feet. because r represents a radius (length), r > 0.
  113. h(0) = 64, so at the moment the object is dropped off the building, the object is 64 feet off of the ground. The solutions to h(t) = 0 are t = \pm 2. because we restrict 0 \leq t \leq 2, we only keep t = 2. This means 2 seconds after the object is dropped off the building, it is 0 feet off the ground. Said differently, the object hits the ground after 2 seconds. The restriction 0 \leq t \leq 2 restricts the time to be between the moment the object is released and the moment it hits the ground.
  114. T(0) = 3, so at 6 AM (0 hours after 6 AM), it is 3^{\circ} Fahrenheit. T(6) = 33, so at noon (6 hours after 6 AM), the temperature is 33^{\circ} Fahrenheit. T(12) = 27, so at 6 PM (12 hours after 6 AM), it is 27^{\circ} Fahrenheit.
  115. C(0) = 27, so to make 0 pens, it costs[1] $ 2700. C(2) = 11, so to make 2000 pens, it costs $1100. C(5) = 2, so to make 5000 pens, it costs $2000.
  116. \item E(0) = 16.00, so in 1980 (0 years after 1980), the average fuel economy of passenger cars in the US was 16.00 miles per gallon. E(14) = 20.81, so in 1994 (14 years after 1980), the average fuel economy of passenger cars in the US was 20.81 miles per gallon. E(28) = 22.64, so in 2008 (28 years after 1980), the average fuel economy of passenger cars in the US was 22.64 miles per gallon.
  117. P(s) = 4s, s > 0.
  118. C(D) = \pi D, D > 0.
  119. .
    1. The amount in the retirement account after 30 years if the monthly payment is $50.
    2. The solution to A(P) = 250000 is what the monthly payment needs to be in order to have 250,000 dollars in the retirement account after 30 years.
    3. A(P+50) is how much is in the retirement account in 30 years if 50 dollars is added to the monthly payment P. A(P)+50 represents the amount of money in the retirement account after 30 years if P dollars is invested each month plus an additional 50 dollars. A(P)+A(50) is the sum of money from two retirement accounts after 30 years: one with monthly payment P dollars and one with monthly payment 50 dollars.
  120. .
    1. because noon is 4 hours after 8 AM, P(4) gives the chance of precipitation at noon.
    2. We would need to solve P(t) \geq 50 \% or P(t) \geq 0.5.
  121. The graph in question passes the horizontal line test meaning for each w there is only one v. The domain of g is [0, \infty) (which is the range of f) and the range of g is [2, \infty) which is the domain of f.
  122. Answers vary.

Section 1.3 Answers

  1. y+1 = 3(x-3) and y = 3x-10
  2. y-8 = -2(x+5) and y = -2x-2
  3. y + 1 = -(x+7) and y = -x-8
  4. y - 1 = \frac{2}{3} (x+2) and y = \frac{2}{3} x + \frac{7}{3}
  5. y - 4 = -\frac{1}{5} (x-10) and y = -\frac{1}{5} x + 6
  6. y - 4 = \frac{1}{7}(x + 1) and y = \frac{1}{7}x + \frac{29}{7}
  7. y - 117 = 0 and y = 117
  8. y + 3 = -\sqrt{2}(x - 0) and y = -\sqrt{2}x - 3
  9. y - 2\sqrt{3} = -5(x - \sqrt{3}) and y = -5x + 7\sqrt{3}
  10. y + 12 = 678(x + 1) and  y = 678x + 666
  11. y = -\frac{5}{3}x
  12. y = -2
  13. y = \frac{8}{5}x - 8
  14. y = \frac{9}{4}x - \frac{47}{4}
  15. y = 5
  16. y = -8
  17. y = -\frac{5}{4} x + \frac{11}{8}
  18. y = 2x + \frac{13}{6}
  19. y = -x
  20. y = \frac{\sqrt{3}}{3} x
  21. y =2x-1, slope: m = 2, y-intercept: (0,-1), x-intercept: \left(\frac{1}{2}, 0 \right)
    A line on the cartesian plane. The line intersects the horizontal axis at x=0.5 and the vertical axis at y = -1.
  22. y =3-x, slope: m = -1, y-intercept: (0,3), x-intercept: (3, 0)
    A line on the cartesian plane. The line is decreasing left to right. The line intersects the horizontal axis at x=3 and vertical axis at y= 3.
  23. y = 3, slope: m =0, y-intercept: (0,3), x-intercept: none
    A line on the cartesian plane the line is horizontal at y = 3.
  24. y = 0, slope: m =0, y-intercept: (0,0), x-intercept: \{ (x,0) \, | \, x \text{ is a real number} \}
    A horizontal line on the cartesian plane. The line runs along the x-axis.
  25. y = \frac{2}{3} x + \frac{1}{3}, slope: m = \frac{2}{3}, y-intercept: \left(0, \frac{1}{3}\right), x-intercept: \left(-\frac{1}{2}, 0\right)
    A line on the cartesian plane. The line increases left to right and intersects the horizontal axis at x=-.5 and the horizontal axis at y = 1/3.
  26. y = \dfrac{1-x}{2}, slope: m = -\frac{1}{2}, y-intercept: \left(0, \frac{1}{2}\right), x-intercept: \left(1, 0\right)
    A line which decreases from left to right. The line intersects the horizontal axis at x = 1 and intersects the vertical axis at y =0.5.
  27. w = -\frac{3}{2} v + 3, slope: m = -\frac{3}{2}, w-intercept: \left(0, 3\right), v-intercept: \left(2, 0\right)
    A line on the cartesian plane. The line is decreasing left to right. The line intersects the vertical axis at y = 3 and intersects the horizontal axis at x = 2.
  28. v = -\frac{2}{3} w + 2, slope: m = -\frac{2}{3}, v-intercept: \left(0,2 \right), w-intercept: \left(3,0\right)
    A line on the cartesian plane. The line is decreasing left to right and intersects the vertical axis at y =2 and the horizontal axis at x = 3.
  29. (-1,-1) and \left(\frac{11}{5}, \frac{27}{5}\right)
  30. y = 3x
  31. y = -6x + 20
  32. y = \frac{2}{3} x - 4
  33. y = -\frac{1}{3} x - \frac{2}{3}
  34. y=-2
  35. x=-5
  36. y = -3x
  37. y = \frac{1}{6}x + \frac{3}{2}
  38. y = -\frac{3}{2} x +9
  39. y = 3x-4
  40. x=3
  41. y=0
  42. f(x) =2x-1, slope: m = 2, y-intercept: (0,-1), x-intercept: \left(\frac{1}{2}, 0 \right)
    A line on the cartesian plane. The line intersects the horizontal axis at x=0.5 and the vertical axis at y = -1.
  43. g(t) =3-t, slope: m = -1, y-intercept: (0,3), t-intercept: (3, 0)
    A line on the cartesian plane. The line is decreasing left to right. The line intersects the horizontal axis at x=3 and vertical axis at y= 3.
  44. F(w) = 3, slope: m =0, y-intercept: (0,3), w-intercept: none
    A line on the cartesian plane the line is horizontal at y = 3.
  45. G(s) = 0, slope: m =0, y-intercept: (0,0), s-intercept: \{ (s,0) \, | \, s\text{ is a real number} \}
    A horizontal line on the cartesian plane. The line runs along the x-axis.
  46. h(t) = \frac{2}{3} x + \frac{1}{3}, slope: m = \frac{2}{3}, y-intercept: \left(0, \frac{1}{3}\right), t-intercept: \left(-\frac{1}{2}, 0\right)
    A line on the cartesian plane. The line increases left to right and intersects the horizontal axis at x=-.5 and the horizontal axis at y = 1/3.
  47. h(t) = \frac{2}{3} x + \frac{1}{3}, slope: m = \frac{2}{3}, y-intercept: \left(0, \frac{1}{3}\right), t-intercept: \left(-\frac{1}{2}, 0\right)
    A line which decreases from left to right. The line intersects the horizontal axis at x = 1 and intersects the vertical axis at y =0.5.
  48. domain: (-\infty, \infty), range: [1, \infty), y-intercept: (0,4), x-intercept: none
    The graph of a piecewise function on the cartesian plane. The first piece is a line decreasing from when x is negative infinity to the dot (3,1). The horizontal line is the second piece starts with an open circle at (3,2) and continues at y = 2 as x increases.
  49. domain: (-\infty, \infty), range: [0, \infty), y-intercept: (0,2), x-intercept: (2,0)
    The graph of a piecewise function. The left piece is a line decreasing to the point (2,0), while crossing the horizontal axis at y = 2. The second piece is a line that increases from the point (2,0).
  50. domain: (-\infty, \infty), range: (-4, \infty), y-intercept: (0,0), t-intercepts: (-2,0), (0,0)
    A piecewise function graphed on a cartesian plane. The left piece is a line which decreases left to right. It intersects the horizontal axis at x = -2 and stops at the open circle (0,-4). The second piece starts at the dot (0,0) and increases left to right.
  51. domain: (-\infty, \infty), range: [-3, 3], y-intercept: (0,-3), t-intercept: \left(\frac{3}{2}, 0 \right) = (1.5,0)
    The graph of a piecewise function. The first piece is a horizontal line at y=-3 that stops at (0,-3). The second piece is an increasing line segment that crosses the horizontal axis at x = 3/2 and stops at (3,3). The last piece is another horizontal line at y = 3.
  52. .
    1. A piecewise function on a cartesian plane. The left piece is a horizontal line running along the negative portion of the x-axis and stops at an open circle at (0,0). The second piece begins at the point (0,1) at a closed dot and moves horizontally as x increases.
    2. domain: (-\infty, \infty), range: \{ 0, 1\}
    3. U is constant on (-\infty, 0) and [0, \infty).
    4. U(t-2) = \left\{ \begin{array}{cc} 0 & \text{if } t<2, \\ 1 & \text{if } t \geq 2 \\ \end{array} \right.
      A piecewise function on a cartesian plane. The left piece is a horizontal line running along the negative portion of the x-axis and stops at an open circle at (2,0). The second piece begins at the point (2,1) at a closed dot and moves horizontally as x increases.
  53. f(x) = -3

 

  1. This is called the `fixed' or `start-up' cost. We'll revisit this concept in Example 1.3.8 in Section 1.3.1. ↵

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Functions, Trigonometry, and Systems of Equations Copyright © by Vanessa Coffelt is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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