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Mathematics LibreTexts

1.1.1E: Functions and Function Notation

  • Page ID
    30238
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    Exercise 1.1

    1. The amount of garbage, G, produced by a city with population p is given by \(G = f(p)\). G is measured in tons per week, and \(p\) is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\).
      b. Explain the meaning of the statement \(f(5)=2\).
    2. The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by \(D=g(a)\). a. A garden with area 5000 ft \({}^{2}\) requires 50 cubic yards of dirt. Express this information in terms of the function \(g\).
      b. Explain the meaning of the statement \(g(100)=1\).
    3. Let \(f(t)\) be the number of ducks in a lake t years after 1990. Explain the meaning of each statement:

      a. \(f\left(5\right)=30\) b. \(f\left(10\right)=40\)
    4. Let \(h(t)\) be the height above ground, in feet, of a rocket t seconds after launching. Explain the meaning of each statement:

      a. \(h(1)=200\) b. \(h(2)=350\)
    5. Select all of the following graphs which represent \(y\) as a function of \(x\).
      a屏幕快照 2019-05-30 下午3.14.34.png b屏幕快照 2019-05-30 下午3.15.13.png c屏幕快照 2019-05-30 下午3.15.50.png
      d屏幕快照 2019-05-30 下午3.16.50.png e屏幕快照 2019-05-30 下午3.17.48.png f 屏幕快照 2019-05-30 下午3.18.13.png
    6. Select all of the following graphs which represent \(y\) as a function of \(x\).
      a屏幕快照 2019-05-30 下午3.19.39.png b 屏幕快照 2019-05-30 下午3.20.50.pngc 屏幕快照 2019-05-30 下午3.21.47.png
      d屏幕快照 2019-05-30 下午3.22.25.png e屏幕快照 2019-05-30 下午3.23.00.png f 屏幕快照 2019-05-30 下午3.23.38.png
    7. Select all of the following tables which represent \(y\) as a function of \(x\).
      屏幕快照 2019-05-30 下午3.26.38.png
    8. Select all of the following tables which represent \(y\) as a function of \(x\).
      屏幕快照 2019-05-30 下午3.27.08.png
    9. Select all of the following tables which represent \(y\) as a function of \(x\).
      屏幕快照 2019-05-30 下午3.27.41.png
    10. Select all of the following tables which represent \(y\) as a function of \(x\).
      屏幕快照 2019-05-30 下午3.28.20.png
    11. Select all of the following tables which represent \(y\) as a function of \(x\) and are one-to-one.
      屏幕快照 2019-05-30 下午3.29.11.png
    12. Select all of the following tables which represent \(y\) as a function of \(x\) and are one-to-one.
    13. Select all of the following graphs which are one-to-one functions.
      a. 屏幕快照 2019-05-30 下午3.30.53.png b.屏幕快照 2019-05-30 下午3.31.22.png c. 屏幕快照 2019-05-30 下午3.31.57.png
      d .屏幕快照 2019-05-30 下午3.32.44.png e . 屏幕快照 2019-05-30 下午3.33.16.png f. 屏幕快照 2019-05-30 下午3.33.54.png
    14. Select all of the following graphs which are one-to-one functions.
      a 屏幕快照 2019-05-30 下午3.35.14.png b屏幕快照 2019-05-30 下午3.35.48.png c 屏幕快照 2019-05-30 下午3.36.21.png
      d 屏幕快照 2019-05-30 下午3.37.24.png e 屏幕快照 2019-05-30 下午3.38.22.png f 屏幕快照 2019-05-30 下午3.38.53.png
    15. Given function \(f(x)\) graphed, evaluate \(f(1)\) and \(f(3)\)
      屏幕快照 2019-05-30 下午3.40.57.png
    16. Given function \(f(x)\) graphed, evaluate \(f(1)\) and \(f(3)\)
      屏幕快照 2019-05-30 下午3.41.27.png
    17. Given the function \(g(x)\) graphed here,
      a. Evaluate \(g(2)\)
      b. Solve \(g(x) = 2\)
      屏幕快照 2019-05-30 下午3.42.57.png
    18. Given the function \(f(x)\) graphed here,
      a. Evaluate \(f(4)\)
      b. Solve \(f(x) = 4\)
      屏幕快照 2019-05-30 下午3.44.17.png
    19. Based on the table below,
      a. Evaluate \(f(3)\) b. Solve \(f(x) = 1\)
      \(x\) 0 1 2 3 4 5 6 7 8 9
      \(f(x)\) 74 28 1 53 56 3 36 45 14 47
    20. Based on the table below,
      a. Evaluate \(f(8)\) b. Solve \(f(x) = 1\)
      \(x\) 0 1 2 3 4 5 6 7 8 9
      \(f(x)\) 62 8 7 38 86 73 70 39 75 34


      For 21-34 functions, evaluate: \(f(-2)\), \(f(-1)\), \(f(0)\), \(f(1)\), and \(f(2)\)

    21. \(f(x) = 4 - 2x\)
    22. \(f(x) = 8 - 3x\)
    23. \(f(x) = 8x^2 - 7x + 3\)
    24. \(f(x) = 6x^2 - 7x + 4\)
    25. \(f(x) = -x^3 + 2x\)
    26. \(f(x) = 5x^4 + x^2\)
    27. \(f(x) = 3 + \sqrt{x + 3}\)
    28. \(f(x) = 4 - \sqrt[3]{x - 2}\)
    29. \(f(x) = (x - 2)(x + 3)\)
    30. \(f(x) = (x + 3) (x - 1)^2\)
    31. \(f(x) = \dfrac{x - 3}{x + 1}\)
    32. \(f(x) = \dfrac{x - 2}{x + 2}\)
    33. \(f(x) = 2^x\)
    34. \(f(x) = 3^x\)
    35. Suppose \(f(x) = x^2 + 8x - 4\). Compute the following:

      a. \(f(-1) + f(1)\)
      b. \(f(-1) - f(1)\)
    36. Suppose \(f(x) = x^2 + x + 3\). Compute the following:

      a. \(f(-2) + f(4)\)
      b. \(f(-2) - f(4)\)
    37. Let \(f(t) = 3t + 5\)

      a. Evaluate \(f(0)\)
      b. Solve \(f(t) = 0\)
    38. Let \(g(p) = 6 - 2p\)

      a. Evaluate \(g(0)\)
      b. Solve \(g(p) = 0\)
    39. Match each function name with its equation.
      a. \(y=x\) i. Cube root
      b. \(y=x^{3}\) ii. Reciprocal
      c. \(y=\sqrt[3]{x}\) iii. Linear
      d. \(y=\dfrac{1}{x}\) iv. Square Root
      e. \(y=x^{2}\) v. Absolute Value
      f. \(y=\sqrt{x}\) vi. Quadratic
      g. \(y = |x|\)
      h. \(y = \dfrac{1}{x^2}\)
    40. Match each graph with its equation.
      a. \(y=x\)
      b. \(y=x^{3}\)
      c. \(y=\sqrt[3]{x}\)
      d. \(y=\dfrac{1}{x}\)
      e. \(y=x^{2}\)
      f. \(y=\sqrt{x}\)
      g. \(y = |x|\)
      h. \(y = \dfrac{1}{x^2}\)
      屏幕快照 2019-05-30 下午4.12.47.png
    41. Match each table with its equation.

      a. \(y=x^{2}\)
      b. \(y=x\)
      c. \(y=\sqrt{x}\)
      d. \(y=1/x\)
      e. \(y=|x|\)
      f. \(y=x^{3}\)
      屏幕快照 2019-05-30 下午4.14.16.png
    42. Match each equation with its table

      a. Quadratic
      b. Absolute Value
      c. Square Root
      d. Linear
      e. Cubic
      f. Reciprocal
      屏幕快照 2019-05-30 下午4.20.21.png
    43. Write the equation of the circle centered at (3, -9) with radius 6.
    44. Write the equation of the circle centered at (9, -8) with radius 11.
    45. Sketch a reasonable graph for each of the following functions. [UW] a. Height of a person depending on age.
      b. Height of the top of your head as you jump on a pogo stick for 5 seconds.
      c. The amount of postage you must put on a first class letter, depending on the weight of the letter.
    46. Sketch a reasonable graph for each of the following functions. [UW]

      a. Distance of your big toe from the ground as you ride your bike for 10 seconds.
      b. Your height above the water level in a swimming pool after you dive off the high board.
      c. The percentage of dates and names you’ll remember for a history test, depending on the time you study.
    47. Using the graph shown,

      a. Evaluate \(f(c)\)
      b. Solve \(f(x)=p\)
      c. Suppose \(f(b)=z\). Find \(f(z)\)
      d. What are the coordinates of points \(L\) and \(K\)?
      屏幕快照 2019-05-30 下午4.26.58.png
    48. Dave leaves his office in Padelford Hall on his way to teach in Gould Hall. Below are several different scenarios. In each case, sketch a plausible (reasonable) graph of the function \(s = d(t)\) which keeps track of Dave’s distance s from Padelford Hall at time \(t\). Take distance units to be “feet” and time units to be “minutes.” Assume Dave’s path to Gould Hall is long a straight line which is 2400 feet long. [UW]
      屏幕快照 2019-05-30 下午4.29.51.png
      a. Dave leaves Padelford Hall and walks at a constant spend until he reaches Gould Hall 10 minutes later.
      b. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute. He then continues on to Gould Hall at the same constant speed he had when he originally left Padelford Hall.
      c. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute to figure out where he is. Dave then continues on to Gould Hall at twice the constant speed he had when he originally left Padelford Hall.
      d. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute to figure out where he is. Dave is totally lost, so he simply heads back to his office, walking the same constant speed he had when he originally left Padelford Hall.
      e. Dave leaves Padelford heading for Gould Hall at the same instant Angela leaves Gould Hall heading for Padelford Hall. Both walk at a constant speed, but Angela walks twice as fast as Dave. Indicate a plot of “distance from Padelford” vs. “time” for the both Angela and Dave.
      f. Suppose you want to sketch the graph of a new function \(s = g(t)\) that keeps track of Dave’s distance s from Gould Hall at time t. How would your graphs change in (a) - (e)?
    Answer

    1. a. \(f(40) = 13\)
    b. 2 Tons of garbage per week i s produced by a city with a population of 5000.

    3. a. In 1995 there are 30 ducks in the lake
    b. In 2000 there are 40 ducks in the lake

    5. a, b, d, e

    7. a, b

    9. a, b, d

    11. b

    13. b, c, e, f

    15. \(f(1) = 1\), \(f(3) = 1\)

    17. \(g(2) = 4\), \(g(-3) = 2\)

    19. \(f(3) = 53\), \(f(2) = 1\)

      \(f(-2)\) \(f(-1)\) \(f(0)\) \(f(1)\) \(f(2)\)
    21. 8 6 4 2 0
    23. 49 18 3 4 21
    25. 4 -1 0 1 -4
    27. 4 4.414 4.732 5 5.236
    29. -4 -6 -6 -4 0
    31. 5 DNE -3 -1 -1/3
    33. 1/4 1/2 1 2 4

    35. a. -6
    b. -16

    37. a. 5
    b. \(-\dfrac{5}{3}\)

    39. a. iii
    b. viii
    d. ii
    e. vi
    f. iv
    g. v
    h. vii

    41. a. iv
    b. ii
    c. v
    d. I
    e. vi
    f. iii

    43. \((x - 3)^2 + (y + 9)^2 = 36\)

    45. a. 屏幕快照 2019-08-09 上午10.43.44.pngb. 屏幕快照 2019-08-09 上午10.45.02.pngc. 屏幕快照 2019-08-09 上午10.45.52.png

    47. a. t
    b. a
    c. r
    d. L: (c, t) and K: (a, p)