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This is a lesson on inverse functions and the horizontal line test





Title – Inverse Functions
By – B. DeComo
Primary Subject – Math
Grade Level – 9

PA Academic Standards:

      2.1.8.G Use the inverse relationships between addition, subtraction, multiplication, division, exponentiation and root extraction to determine unknown quantities in equations.
      2.8.8.G Represent relationships with tables or graphs in the coordinate plane and verbal or symbolic rules.
      2.8.8.H Graph a linear function from a rule or table.
    2.8.8.J Show that an equality relationship between two quantities remains the same as long as the same change is made to both quantities; explain how a change in one quantity determines another quantity in a functional relationship.

Goal of this lesson:

  • To develop understanding of inverse functions and inverse relations
  • To identify and find inverse relations and inverse functions
  • To identify real world situations that inverse functions are used
  • To determine from a graph if the inverse of a given function is a function using the horizontal line test


  • Paper / Graph paper
  • Chalk
  • Worksheet / Answer key
  • Transparencies / Blank transparencies
  • Introduction review sheet / answer key
  • 5 decks of playing cards

Clerical/Administrative Tasks

  • Take roll
  • Make activity sheet/ answer key
  • Make copies of activity sheet/ review sheet
  • Make review sheet/ answer key
  • Make transparencies

Instructional Objectives

  • TSWBAT state the definition of an inverse relation and inverse function
  • TSWBAT identify and recognize inverse relations and inverse functions through examples
  • TSWBAT use and identify inverse functions in real life situations
  • TSWBAT determine from a graph if the inverse of a given function is a function using the horizontal line test

Key terms:

      Inverse Function

      Inverse Relation


    Horizontal line test

Introduction (3-5 minutes)

  • Go over three homework problems from previous day that students had most difficulty with. (use homework tally sheet to choose problems) (5 minutes)
  • Discuss example of function and inverse function about wrapping / unwrapping a gift.

Developmental Activities (15-20 minutes)

  • So today class, we are going to learn about inverse relations and inverse functions
  • Pass out class starter worksheet about converting Fahrenheit to Celsius degrees. (Transparency 1)
    Give students 3-4 minutes to do this sheet (MBWA)
    Go over sheet in class
  • Discuss what an inverse relation is and how to obtain it from a set of ordered pairs
  • Show and give examples of how to find an inverse relation from the original relation
  • Give example for students to find inverse relation from original relation
  • Discuss what a function is
    Remember that in order for a relation to be a function it must pass the vertical line test.
  • Give different examples about relations and inverse relations.
    Both relation and inverse relation are functions (show mapping)
    Relation is a function but inverse relation is not a function (show mapping)
    Give example for students to do
  • Plot points on same graph to show how relation and inverse relation points form a symmetrical arrangement
  • Discuss that the graph of the inverse is the reflection of the graph of the original relation.
  • Show solution / problem reflection
  • Discuss and show psych picture.
  • Ask students to look at picture and tell me what they see
  • Symmetric along y=x
  • Give examples of a function and its inverse on same graph
  • Discuss horizontal line test and how it determines if the inverse is a function as well.
  • Horizontal line test: the inverse of a function is a function if and only if every horizontal line intersects the graph of the given function at no more than one point.
  • Show examples and non examples of inverse functions on graphs using horizontal line test
  • Start to explain how to find the inverse of a function mathematically. (This will lead into what will be discussed and taught tomorrow; start only if time permits)
  • Use function, y = 3x + 1. (Follow teaching notes/ transparencies).

Assessment/Evaluation (10 minutes)

  • Worksheet on inverses of functions using cooperative learning strategy worksheet checkmates
  • State directions, pass out worksheet, MBWA

**If time permits activity**

      will be discussed more in depth and demonstrated during class. Break students into 5 groups. Give each group a deck of playing cards. Each student will draw two cards at a time for a total of 5 times. This will give each student 10 cards. Each two cards they draw will represent an ordered pair (relation). Graph relation and inverse relation on graph paper. Determine if a relation and inverse relation are functions using vertical and horizontal line tests.

      1=1, 2=2, 3=3, … 10=10,

      jack=11, queen=12, king=13

      negative x numbers = spades

      positive x numbers = clubs

      negative y numbers = diamonds

      positive y numbers = hearts
    *What students do not get done is for homework*

Conclusion (3-5 minutes)

  • Tomorrow class we are going to discuss how to find an inverse function mathematically.
  • Students will use the “turn to your neighbor” cooperative learning strategy and explain how to find inverse relations and inverse functions.
  • Students will write at least three facts they learned and how to use inverse functions in real life situations in their journals


      Dilley, Clyde and Meiring, Steven and Tarr, John and Taylor, Ross.

Algebra 2 with Trigonometry

      . Massachusetts. D.C Heath and Company. 1990
      Larson, Roland and Kanold, Timothy and Stiff, Lee.

Algebra 2 an Integrated Approach

      . Massachusetts. D.C. Heath and Company. 1995
      Schultz, James and Ellis, Wade Jr. and Hollowell, Kathleen and Kennedy, Paul.

Algebra 2

    . Austin. Holt, Rinehart, and Winston. 2001.

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