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The Mathematics of Functions

Unit Plan

The Mathematics of Functions

Objectives

This unit explores functions, from the definition and various representations of functions through the concepts of inverse functions, as well as operations with functions, specifically composition. Students will:

  • learn the definition of a function and distinguish functions from nonfunctions.
  • learn to find the inverse of functions algebraically using a table or graph.
  • compose functions written as equations as well as graphs.

Essential Questions

  • What is a function and why are functions so mathematically interesting and necessary?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Formative Assessment

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    Multiple-Choice Items:

    1. Which of the following graphs is not a function?

     

    2. Based on this table, which of the following statements is true?

     

    x

    y

    1

    3

    2

    1

    3

    5

     

     

    A      f(3) = 7

    B       f(1) = 2

    C       f(1) = 3

    D      f(2) does not exist

     

     

    3. A doctor knows that medicine is most effective immediately after a patient takes it. Then the effectiveness of the medicine decreases. After 6 hours, the doctor gives the patient more medicine. Which graph could model this situation?

     

     

    4. Lulu is sewing a dress. The dress needs 2 feet of ribbon around the waist. Also, Lulu will add bows, each of which requires 0.5 foot of ribbon. Which table correctly models the situation?

     

    A

    Number of Bows

    Total Feet of Ribbon Needed

    1

    3

    2

    5

    3

    7

     

    B

    Number of Bows

    Total Feet of Ribbon Needed

    1

    2

    2

    2

    3

    2

     

    C

    Number of Bows

    Total Feet of Ribbon Needed

    1

    2

    2

    2.5

    3

    3

     

    D

    Number of Bows

    Total Feet of Ribbon Needed

    1

    2.5

    2

    3

    3

    3.5

     


    Use the following graph to answer question 5.

     

     

    5. The graph above models Maria’s bike ride to school. Which situation describes her ride most accurately?

    A      Maria left home, rode for 2 minutes, and then returned home to get her homework. After getting her homework, she finished her bike ride to school.

    B       Maria left school, rode for 3 minutes, rode back to school, and then rode home.

    C       Maria rode straight to school. The entire ride took 2 minutes.

    D      Maria rode her bike for 3 minutes and then stopped to rest for 2 minutes. Then she rode the remaining distance.

     

    6. Which of these tables is not a function?

     

    A

    x

    y

    2

    3

    3

    3

    4

    3

     

     

     

    B

    x

    y

    1

    5

    1

    4

    1

    3

     

     

     

    C

    x

    y

    −2

    4

    0

    0

    2

    4

     

     

     

    D

    x

    y

    5.1

    4.2

    3.9

    5.8

    2.7

    6.1

    Use the graph below to answer question 7.

     

    7. Which of the following tables represents this graph?

     

    A

    x

     f(x)

    −1

    3

    0

    4

    1

    5

     

     

     

    B

    x

    f(x)

    −1

    5

    0

    4

    1

    3

     

     

     

    C

    x

    f(x)

    3

    −1

    4

    0

    5

    1

     

     

     

    D

    x

    f(x)

    −1

    −1.5

    0

    1

    1

    1.5

     

     

    8. How are the graph of a function and the graph of its inverse related?

     

    A      They are reflections of one another across the x-axis.

    B       They are reflections of one another across the y-axis.

    C       They are reflections of one another across the line y = x.

    D      They are reflections of one another across the origin.

     

     

    9. For the functions f(x) = 2x − 5 and g(x) = 3x − 2, what is f(g(x))?

    A.  2x − 5

    B.  6x − 9

    C.  4x − 15

    D.  6x − 17

     

    Multiple-Choice Answer Key:

    1. C

    2. C

    3. B

    4. D

    5. D

    6. B

    7. B

    8. C

    9. B

     

     


    Short-Answer Items:

    10. Find the inverse of  and draw its graph.

     

     

     

     

     

     

     

     

    11. Let f(x) = 2x − 1 and g(x) = x2. Find f(g(2)).

     

     

     

     

     

     

     

    12. Graph . Is this a function? Using the vertical line test, tell why it is or is not a function.


    Short-Answer Key and Scoring Rubrics:

    Remember that f-1(x) means the inverse function of f(x), not f to the negative 1 power. Students may use another letter to represent the inverse function as well, so be prepared for that.

    10. Find the inverse of  and draw its graph.

    The graph for the inverse should look like this, though keep in mind scales and viewing regions may vary:

     

     

     

    Points

    Description

    2

    • Response is complete, correct, and detailed. Student translates between the function and its inverse. The graph is correctly drawn with slope =2 and y-intercept = 6.

    1

    • Response is partially correct, but has no graph or the graph does not represent the inverse.

    0

    • Incorrect response; no graph or incorrect graph.

     

     

     

    11. Let f(x) = 2x − 1 and g(x) = x2. Find f(g(2)).

    f(g(x)) = 2(2)2−1 = 7

     

    Points

    Description

    2

    • Response is complete and correct, showing the calculation.

    1

    • Response is correct, but includes no supporting calculations or path to the solution.

    0

    • Incorrect response; no appropriate supporting work.

     

    12. Graph . Is this a function? Using the vertical line test, tell why it is or is not a function.

    Yes. The relation is a function. The vertical line test shows one and only one output for each input.

     

     

     

    Points

    Description

    2

    • Response is Yes and indicates that the vertical line test shows one and only one output for each input.

    1

    • Response is Yes, but includes no adequate supporting work or path to the solution.

    0

    • Incorrect response; no appropriate supporting work.

     


    Performance Assessment:

    A certain cell phone company charges $0.03 per text message as well as a flat rate of $4 per month. Show your work.

    1. Let T stand for the number of text messages sent and C stand for total cost. Write a linear equation to model this situation where C is a function of T.
    2. Find the inverse of your function from Part 1.
    3. Explain what the inverse means in terms of the situation.

    Performance Assessment Scoring Rubric:

    1. C = 4 + 0.03 * T
    2. The inverse tells you how many texts you can send for a certain monthly cost.

     

    Points

    Description

    4

    • Both equations are correctly represented in terms of C and T.
    • Inverse is expressed as number of texts per cost.
    • Supporting work is complete and correct.

    3

    • Parts 1 and 2, or 2 and 3, or 1 and 3 are correct.
    • Appropriate supporting work is provided for Part 1 or 2.

    2

    • Part 1 or 2 or 3 is correct.
    • Appropriate supporting work that leads to the solution is provided.

    1

    • Part 1 or 2 or 3 is correct.
    • Supporting work is not present.

    0

    • Parts 1, 2, and 3 are incorrect.
    • Supporting work is incorrect, unclear, or not provided.

     

DRAFT 10/12/2011
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