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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
- Worksheet / Answer key
- Transparencies / Blank transparencies
- Introduction review sheet / answer key
- 5 decks of playing cards
- Take roll
- Make activity sheet/ answer key
- Make copies of activity sheet/ review sheet
- Make review sheet/ answer key
- Make transparencies
- 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
- 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.
- . Austin. Holt, Rinehart, and Winston. 2001.
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