More Basic Functions and Graphs: Get Stronger Answer Key

Trigonometric Functions

  1. [latex]\frac{4\pi}{3}[/latex] rad
  2. [latex]\frac{−\pi }{3}[/latex] rad
  3. [latex]\frac{11\pi}{6}[/latex] rad
  4. [latex]210°[/latex]
  5. [latex]-540°[/latex]
  6. [latex]-1/2[/latex]
  7. [latex]-\large\frac{\sqrt{2}}{2}[/latex]
  8. [latex]\large \frac{\sqrt{3}-1}{2\sqrt{2}} \normalsize = \large \frac{\sqrt{6}-\sqrt{2}}{4}[/latex]
    1. [latex]b=5.7[/latex]
    2. [latex]\sin A=\frac{4}{7}, \, \cos A=\frac{5.7}{7}, \, \tan A=\frac{4}{5.7}, \, \csc A=\frac{7}{4}, \, \sec A=\frac{7}{5.7}, \, \cot A=\frac{5.7}{4}[/latex]
    1. [latex]c=151.7[/latex]
    2. [latex]\sin A=0.5623, \, \cos A=0.8273, \, \tan A=0.6797, \, \csc A=1.778, \, \sec A=1.209, \, \cot A=1.471[/latex]
    1. [latex]c=85[/latex]
    2. [latex]\sin A=\frac{84}{85}, \, \cos A=\frac{13}{85}, \, \tan A=\frac{84}{13}, \, \csc A=\frac{85}{84}, \, \sec A=\frac{85}{13}, \, \cot A=\frac{13}{84}[/latex]
    1. [latex]y=\frac{24}{25}[/latex]
    2. [latex]\sin \theta =\frac{24}{25}, \, \cos \theta =\frac{7}{25}, \, \tan \theta =\frac{24}{7}, \, \csc \theta =\frac{25}{24}, \, \sec \theta =\frac{25}{7}, \, \cot \theta =\frac{7}{24}[/latex]
    1. [latex]x=\frac{−\sqrt{2}}{3}[/latex]
    2. [latex]\sin \theta =\frac{\sqrt{7}}{3}, \, \cos \theta =\frac{−\sqrt{2}}{3}, \, \tan \theta =\frac{−\sqrt{14}}{2}, \, \csc \theta =\frac{3\sqrt{7}}{7}, \, \sec \theta =\frac{-3\sqrt{2}}{2}, \, \cot \theta =\frac{−\sqrt{14}}{7}[/latex]
  9. [latex]\sec^2 x[/latex]
  10. [latex]\sin^2 x[/latex]
  11. [latex]\sec^2 \theta[/latex]
  12. [[latex]\large\frac{1}{\sin t} \normalsize =\csc t[/latex]
  17. [[latex]\theta = \{\frac{\pi}{6}, \, \frac{5\pi}{6}\}[/latex]
  18. [latex]\theta = \{\frac{\pi}{4}, \, \frac{3\pi}{4}, \, \frac{5\pi}{4}, \, \frac{7\pi}{4}\}[/latex]
  19. [latex]\theta = \{\frac{2\pi}{3}, \, \frac{5\pi}{3}\}[/latex]
  20. [latex]\theta = \{0, \, \pi, \, \frac{\pi}{3}, \, \frac{5\pi}{3}\}[/latex]
  21. [latex]y=4\sin\Big(\large\frac{\pi}{4} \normalsize x\Big)[/latex]
  22. [latex]y=\cos(2\pi x)[/latex]
    1. [latex]1[/latex]
    2. [latex]2\pi[/latex]
    3. [latex]\frac{\pi}{4}[/latex] units to the right
    1. [latex]\frac{1}{2}[/latex]
    2. [latex]8\pi[/latex]
    3. No phase shift
    1. [latex]3[/latex]
    2. [latex]2[/latex]
    3. [latex]\frac{2}{\pi}[/latex] units to the left
  23. Approximately [latex]42[/latex] in.
    1. [latex]0.550[/latex] rad/sec
    2. [latex]0.236[/latex] rad/sec
    3. [latex]0.698[/latex] rad/min
    4. [latex]1.697[/latex] rad/min
  24. [latex]\approx 30.9 \, \text{in}^2[/latex]
    1. [latex]\pi /184[/latex]; the voltage repeats every [latex]\pi /184[/latex] sec
    2. Approximately [latex]59[/latex] periods
    1. Amplitude = [latex]10[/latex]; period = [latex]24[/latex]
    2. [latex]47.4^{\circ} F[/latex]
    3. [latex]14[/latex] hours later, or 2 p.m.
    4. An image of a graph. The x axis runs from 0 to 365 and is labeled “t, hours after midnight”. The y axis runs from 0 to 20 and is labeled “T, degrees in Fahrenheit”. The graph is of a curved wave function that starts at the approximate point (0, 41.3) and begins decreasing until the point (2, 40). After this point, the function increases until the point (14, 60). After this point, the function begins decreasing again.

Inverse Functions

    1. Not one-to-one
    2. Not one-to-one
    3. One-to-one
      1. [latex]f^{-1}(x)=\sqrt{x+4}[/latex]
      2. Domain: [latex]x \ge -4[/latex], Range: [latex]y \ge 0[/latex]
      1. [latex]f^{-1}(x)=\sqrt[3]{x-1}[/latex]
      2. Domain: all real numbers, Range: all real numbers
      1. [latex]f^{-1}(x)=x^2+1[/latex]
      2. Domain: [latex]x \ge 0[/latex], Range: [latex]y \ge 1[/latex]
    7. An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of two functions. The first function is an increasing straight line function labeled 'f'. The x intercept is at (-2, 0) and y intercept are both at (0, 1). The second function is of an increasing straight line function labeled 'f inverse'. The x intercept is at the point (1, 0) and the y intercept is at the point (0, -2).
    8. An image of a graph. The x axis runs from 0 to 8 and the y axis runs from 0 to 8. The graph is of two functions. The first function is an increasing straight line function labeled 'f'. The function starts at the point (0, 1) and increases in a straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1). The second function is an increasing straight line function labeled 'f inverse'. The function starts at the point (1, 0) and increases in a straight line until the point (6, 4). After this point, the function continues to increase, but at a faster rate than before, as it approaches the point (8, 8). The function does not have a y intercept and the x intercept is (1, 0).
    9. These are inverses.
    10. These are not inverses.
    11. These are inverses.
    12. These are inverses.
    13. [latex]\frac{\pi}{6}[/latex]
    14. [latex]\frac{\pi}{4}[/latex]
    15. [latex]\frac{\pi}{6}[/latex]
    16. [latex]\frac{\sqrt{2}}{2}[/latex]
    17. [latex]-\frac{\pi}{6}[/latex]
      1. [latex]x=f^{-1}(V)=\sqrt{0.04-\frac{V}{500}}[/latex]
      2. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity [latex]V[/latex].
      3. [latex]0.1[/latex] cm; [latex]0.14[/latex] cm; [latex]0.17[/latex] cm
      1. [latex]$31,250, $66,667, $107,143[/latex]
      2. [latex]p=\frac{85C}{C+75}[/latex]
      3. [latex]34[/latex] ppb
      1. [latex]~92^{\circ}[/latex]
      2. [latex]~42^{\circ}[/latex]
      3. [latex]~27^{\circ}[/latex]
    21. [latex]x \approx 6.69, 8.51[/latex]; so, the temperature occurs on June 21 and August 15
    22. [latex]~1.5 sec[/latex]
    23. [latex]\tan^{-1}(\tan(2.1))\approx -1.0416[/latex]; the expression does not equal [latex]2.1[/latex] since [latex]2.1>1.57=\frac{\pi}{2}[/latex]—in other words, it is not in the restricted domain of [latex]\tan x[/latex]. [latex]\cos^{-1}(\cos(2.1))=2.1[/latex], since [latex]2.1[/latex] is in the restricted domain of [latex]\cos x[/latex].

    Exponential and Logarithmic Functions

      1. [latex]125[/latex]
      2. [latex]2.24[/latex]
      3. [latex]9.74[/latex]
      1. [latex]0.01[/latex]
      2. [latex]10,000[/latex]
      3. [latex]46.42[/latex]
    3. b
    4. a
    5. c
    6. Domain: all real numbers, Range: [latex](2,\infty)[/latex], Horizontal asymptote at [latex]y=2[/latex]
    7. Domain: all real numbers, Range: [latex](0,\infty)[/latex], Horizontal asymptote at [latex]y=0[/latex]
    8. Domain: all real numbers, Range: [latex](-\infty ,1)[/latex], Horizontal asymptote at [latex]y=1[/latex]
    9. Domain: all real numbers, Range: [latex](-1,\infty )[/latex], Horizontal asymptote at [latex]y=-1[/latex]
    10. [latex]8^{1/3}=2[/latex]
    11. [latex]5^2=25[/latex]
    12. [latex]e^{-3}=\frac{1}{e^3}[/latex]
    13. [latex]e^0=1[/latex]
    14. [latex]\log_4(\frac{1}{16})=-2[/latex]
    15. [latex]\log_9 1=0[/latex]
    16. [latex]\log_{64} 4=\frac{1}{3}[/latex]
    17. [latex]\log_9 150=y[/latex]
    18. [latex]\log_4 0.125=-\frac{3}{2}[/latex]
    19. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = 1”. There is no y intercept and the x intercept is at the approximate point (2, 0).Domain: [latex](1,\infty )[/latex], Range: [latex](−\infty ,\infty)[/latex], Vertical asymptote at [latex]x=1[/latex]
    20. An image of a graph. The x axis runs from -1 to 9 and the y axis runs from -5 to 5. The graph is of a decreasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (e, 0).Domain: [latex](0,\infty)[/latex], Range: [latex](−\infty ,\infty)[/latex], Vertical asymptote at [latex]x=0[/latex]
    21. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line “x = -1”. There y intercept and the x intercept are both at the origin.Domain: [latex](-1,\infty)[/latex], Range: [latex](−\infty ,\infty)[/latex], Vertical asymptote at [latex]x=-1[/latex]
    22. [latex]2+3\log_3 a-\log_3 b[/latex]
    23. [latex]\frac{3}{2}+\frac{1}{2}\log_5 x+\frac{3}{2}\log_5 y[/latex]
    24. [latex]-\frac{3}{2}+\ln 6[/latex]
    25. [latex]\frac{\ln 15}{3}[/latex]
    26. [latex]\frac{3}{2}[/latex]
    27. [latex]\log 7.21[/latex]
    28. [latex]\frac{2}{3}+\frac{\log 11}{3\log 7}[/latex]
    29. [latex]x=\frac{1}{25}[/latex]
    30. [latex]x=4[/latex]
    31. [latex]x=3[/latex]
    32. [latex]1+\sqrt{5}[/latex]
    33. [latex]\frac{\ln 82}{\ln 7} \approx 2.2646[/latex]
    34. [latex]\frac{\ln 211}{\ln 0.5} \approx -7.7211[/latex]
    35. [latex]\frac{\ln 0.452}{\ln 0.2} \approx 0.4934[/latex]
    36. [latex]\approx 17,491[/latex]
    37. Approximately [latex]$131,653[/latex] is accumulated in [latex]5[/latex] years.
    38. a. [latex]\approx 333[/latex] million b. [latex]94[/latex] years from 2013, or in 2107
      1. a. [latex]k \approx 0.0578[/latex]
      2. b. [latex]\approx 92[/latex] hours
    40. The San Francisco earthquake had [latex]10^{3.4}[/latex] or [latex]\approx 2512[/latex] times more energy than the Japan earthquake.