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Grade Level/Course:Algebra 1 & Algebra 2Lesson/Unit Plan Name:Introduction to Inverse FunctionsRationale/Lesson Abstract:This lesson is designed to introduce inverse functions by building on students’ prior knowledgeof functions and graphing functions. It is intended for students to not only see the two variablesexchange places in the equation, but also to highlight how the characteristics of the two graphsrelate & vary based on this relationship. Students should have a firm grasp of graphing a linearfunction as well as writing the equation of a line in slope intercept form.Timeframe:One periodCommon Core Standard(s):F.BF.4 – Find Inverse FunctionsInstructional Resources/Materials:Warm up, student note-taking guide, graph paper and pencilPage 1 of 10MCC@WCCUSD 10/05/13
Activity/Lesson:Warm Up Solutions:Answers A, C, and D are relations that are functions.Use this problem to remind students about thevertical line test and why the test works – If avertical line hits two or more points, then therelation has an input that provides two or moreoutputs and is not a function.a.) Function – A relation which associates everyelement in one set (the domain) with oneand only one element in another set (therange).Domain: -3, 0, 1, 2, 5Range: -3, 1, 4, 81y 7 x21y 7 7 x 72b.) Yes. Every element in the domain has onlyone element in the range.1y x 7211y y x 7 x 7 22y 2 x 14Introduction to Inverse Functions:Pass out student note-taking guide and begin by defining the inverse of a function:“The inverse of a function is the relation formed when the independent variable is exchanged with the dependentvariable of a given relation.”Give this example to make sure students understand the definition of an inverse.Example 1:Find the inverse of the following function:f : 3,4 , 2,8 , 1,7 , 0, 3 Answer: f 1: 4,3 , 8,2 , 7, 1 , 3,0 Let students know that fexponent. 1is called “the inverse of f ” or “ f inverse” and also let them know that the -1 is not anPage 2 of 10MCC@WCCUSD 10/05/13
Example 2:Graph the function and its inverse. Then find the equation of the inverse function.f ( x) 2x 6x x 1m y 1 80 61 42 2 y 2 y 2 2 x 18-8y8-8xHave students fill out the tablefor the original function withthe following inputs at left andgraph the points on the graphas you go. Then remindstudents that the slope can beseen from the table by findingthe ratio of the change in yover the respective change in x.Then have students exchangethe x and y coordinates of yourtable(left) to form points onyour inverse function(table onright) just like example 1. Havethe students graph these pointsas well and then draw theinverse function. Continue thepattern in the table and on thegraph to find the y-intecept ofthe inverse function and thenwrite the equation of theinverse function.f 1( x)xy 8 1 6 0 x 2m 41 22 y 1 y 1 1 x 2 28y-88x-8The y-intercept is 0,3 b 3Inverse: y 1x 32f 1 x 1x 32Think-Pair-ShareHave the students compare the two graphs individually and look for any relationships/differences between the two. Thenask them to share with a partner. After a few minutes have students share out what they noticed. Write conjectures forall to see.Page 3 of 10MCC@WCCUSD 10/05/13
Example 2 continued:Finding the Inverse FunctionAlgebraicallyf ( x) 2 x 6y 2x 6f 1: x 2y 6x 6 2y 6 6x 6 2yx 6 2y 22x 6 y2 2Before this step, ask a student to rereadthe definition of an inverse. Thenexchange the two variables in theequation and let the students know thatthe solutions for the equation are nowreversed. For example in the originalfunction an x-value of 3 produces a yvalue of 0 (3,0), while in the inverserelation a y-value of 3 produces an x-valueof 0 (0,3). Solving for y as a function of xcreates your inverse function.1x 3 y2 f 1 x 1 x 32You Try:2Find the inverse of the function f x x 4 .3Walk around the room and find a student to debrief their workafter giving everyone a few minutes to work it out. If anotherstudent did it graphically, have them show that as well and seeif some of the relationships/differences discussed earlier duringthe Think-Pair-Share hold true.f 1f ( x) 2x 43y 2x 4323: x y 4x 4 2y 4 43x 4 2y3 2 3 x 4 3 y 3 3 x 12 2 y3 x 12 2 y 2 2 3x 6 y2 fPage 4 of 10 1 x 3 x 62MCC@WCCUSD 10/05/13
Example 3:What is the inverse of the function whose graph is shown below? 8Label the two pointsexaggerated on the graphof the function.Exchange their coordinatesto find two points of theinverse function.Plot the two points andgraph the inverse functionby drawing the line throughthe two points.Then find the slope and yintercept of the inverse towrite the equation of theinverse function. y8y f ( x)y f ( x) 0,2 6,0 -88y 0,6 0,2 x 6,0 2,0 -8-8m xy f 1 ( x)-8f 1 :8riserunb 6 6 2 3 f 1 x 3 x 6Think-Pair-ShareIs there another way we could have used to find the inverse in example 3? Give the students a minute to thinkabout it. Then ask them to share with a partner. Discuss as a class. Then have them verify that the inversefunction is correct by finding the equation of f ( x) and then finding the inverse of f algebraically:8y1f x x 23y f ( x) 0,2 6,0 -881y x 23xf 1 :1x 2 y 2 23-8f :risem run 1x 2 x3b 2 1 3 x 2 3 y 3 2 6 1x y 23 3 x 2 y13 3 x 6 y1 f x x 23 f 1 x 3 x 6Page 5 of 10MCC@WCCUSD 10/05/13
You TryDraw the graph of the inverse function for each of the functions shown below on the same coordinate plane.a.)b.)c.)yyy8-88x-8-88888x-88-8-8y xyy f ( x) yxy x88yy xy f ( x)y f ( x)-88x-88xy f 1 ( x)-88y f 1 ( x)y f 1 ( x)-8-8-8After giving the students a few minutes to finish the you try, draw the inverse functions on each of the graphs.Draw the line y x on all three coordinate planes. Let the students know that “the inverse relation is areflection of the original function across the line y x ”. Introduce this relationship by focusing on c.) toexplain what a reflection is and show how every point on the graph of the relation can be reflected across theline y x to form the graph of its inverse.Page 6 of 10MCC@WCCUSD 10/05/13x
Assessment:Have students think about what they learned today. Wait a minute then randomly select students to piecetogether a summary representative of the material discussed: The inverse of a function is formed when you exchange the independent and dependent variables of agiven relation You can find the inverse function from the graph of a line by exchanging the x and y coordinates of theintercepts, plotting these new points, drawing the line and then writing the equation of the line. You can find the inverse from the equation of a function by replacing f ( x) with y, exchanging the xand y variables in the equation, solving for y, and then replacing y with f 1 ( x) .Exit Ticket –1) Find the inverse of the function graphed below:8-812) Find the inverse of the function f ( x) x 103y8x-8Page 7 of 10MCC@WCCUSD 10/05/13
Warm-UpCST/CCSS: 18.0/F.IF.1Select all the following relations thatare functions:A)B)C)D)Review: Algebra 1 F.IF.1yFind the domain and range of the relation: 5,4 , 0,4 , 3,1 , 2,8 , 1, 3 xCurrent: Algebra 1 F.IF.1a) What is a function?Other: Algebra 1 A.CED.4Solve for y.1y 7 x2b) Is the relation in quadrant I a function?Page 8 of 10MCC@WCCUSD 10/05/13
Introduction to Inverse Functions: Note-Taking GuideDefinition of the inverse of a function:Example 1: Find the inverse of the following function f : 3,4 , 2,8 , 1,7 , 0, 3 .Example 2: Graph the function f ( x) 2x 6 and its inverse. Then find the equation of the inverse function.f ( x) 2x 6x8-8fyxy88-8x 1 x Finding the Inverse FunctionAlgebraicallyyy-88x-8Page 9 of 10MCC@WCCUSD 10/05/13
You Try:2Find the inverse of the function f x x 4 .3y8-8x8-8Example 3:What is the inverse of the function whose graph is shown below?y8-88x-8You TryDraw the graph of the inverse function for each of the functions showed below on the same coordinate plane.a.)b.)c.)yyy8-88-888x-88-8Page 10 of 10x-88-8MCC@WCCUSD 10/05/13x
Algebra 1 & Algebra 2 Lesson/Unit Plan Name: Introduction to Inverse Functions . Common Core Standard(s): F.BF.4 - Find Inverse Functions Instructional Resources/Materials: Warm up, student note-taking guide, graph paper and pencil . Page 2 of 10 MCC@WCCUSD 10/05/13
