Intro To Inverse Functions - About

Intro to InverseFunctionsReturn toTable ofContents123

Goals and ObjectivesStudents will be able to recognize and find an inverse function:a) using coordinates,b) graphically andc) algebraically.124

Why Do We Need This?Sometimes, it is important to look at a problem from the inside out.Addition undoes subtraction.Multiplication undoes division.In order to look deeply into different problems,we must try to see things from the inside out.Inverse functions undo original functions.125

Inverse FunctionAn inverse function is a function that undoes the action of anotherfunction. The inverse function has all of the same points as theoriginal function, except the domain and range values (or x and yvalues) have been switched. In other words, the domain of onefunction will be the range of its inverse & vice versa.The notation for the inverse ofis.Read, "the inverse of f of x."You can prove that a function is an inverse of another using thefollowing relationship:126

Inverse Functionare inverse functions.AnswerProve thatMa127

Inverse FunctionYou can also prove that two functions are inverses by GRAPHING.The graph of an inverse is the reflection of the function over the line.What conjecture can youmake about the x and yvalues of inverse functions?Explain your answer.AnswerThe following functions areinverses of each other.Thswitche.g.whichinter128

Inverse FunctionThe inverse of a function is the reflection over the line. As youobserved, this will result in the switching of x and y values.Examples:b) Find the inverse of:32445–567Answera) Find the inverse of:a)b)129

64 What is the inverse of?BCAnswerAD130

65 If the inverse of a function iswhat was the original function?,BCAnswerAD131

66 What is the inverse of:A434334100110233 2 23477447AnswerBDC 3 443 10002 3 23 4 774132

67 Will the inverse of the following points be a function?Why or why not?No12233452AnswerYesNo. Ifvalue2 wiU133

InverseDraw the inverse of the given function.AnswerFunctioLinfunctioIs the inverse a function? Why or why not?134

InverseAnswerDraw the inverse of the given function.Is the inverse a function? Why or why not?135

Horizontal Line TestThe Horizontal Line Test is used to determine if the inverseof a function is also considered to be a function. If ahorizontal line crosses the function more than once, itsinverse is NOT a function.Is the inverse of thefunction shown to theright considered to be afunction?AnswerJust like the Vertical Line Test, there is a simple way to determine if afunction's inverse is also a function, just from looking at its graph.Thecheckiis a fuhas anecThisMove this line to check:136

One to one FunctionsFunctions are one to one when every element in the domain ismapped, or connected to, a unique element in the range.This happens when the original function passes both the verticaland horizontal line test.An example of two one to one functions are shown in the graphsbelow.137

Horizontal Line TestExample: Will the inverse of the given function be a function?AnswerMove this line to check:138

Answer68 Which graph is the inverse of the function below?ABCD139

Answer69 Which graph is the inverse of the function below?ABCD140

Answer70 Which graph is the inverse of the function below?ABCD141

71 Will the inverse ofbe a function?YesAnswerNo142

Finding the Inverse of aFunction AlgebraicallyKnowing that the inverse of a function switches x and y values,we can take this concept further when given an equation.Given:ChangeSwitchtoandSolve forChange it back to function notation, using, since it's the inverseNote: If the original equation is written as "", it's inverse is143

InversePractice: Find the inverse of the following functions.a)b)Answera)144

InversePractice: Find the inverse of the following function.Answerc)145

AB?Answer72 Which of the following choices is the inverse ofCD146

73 Which of the following is the inverse of?ABCUse thisthem ofMost stuDAnswerBanswer147

74 Find the inverse ofBCAnswerAD148

Applications of InversesPart ATwo children sit on a seesaw, as illustrated. The mass, inkilograms, of the first child isand the mass, in kilograms, of thesecond child is. In the diagram,represent thedistance, in meters, from the fulcrum (the balance point) to eachchild. The total distance between the children is 5 meters.For a seesaw to be balanced,. Use the information inthe in the table to write the functionthat allows you todetermine, the mass of the second child.45x5 xPart BDetermine the inverse functionto model the distance,based on the mass of the first child. Show your work.,149

Applications of InversesPart ATwo children sit on a seesaw, as illustrated. The mass, inkilograms, of the first child isand the mass, in kilograms, of thesecond child is. In the diagram,represent thedistance, in meters, from the fulcrum (the balance point) to eachchild. The total distance between the children is 5 meters.For a seesaw to be balanced,. Use the information inthe in the table to write the functionthat allows you todetermine, the mass of the second child.45x5 xTo start, use the balance equation provided, and fill in the knownvalues & expressions.150

Applications of InversesNow, solve the equation foras the function. In the final step, write your equation151

Applications of InversesPart BDetermine the inverse functionto model the distance,based on the mass of the first child. Show your work.,152

Use the information provided to write the functionthat allowsyou to determine , Audra's speed during her trip from the officebuilding to her house.ACBDAnswer75 Part A: Every morning and evening, Audra needs to commutebetween her house to the office building. When traveling from herhouse to the office building, she leaves early enough that she cantravel at a regular highway speed of 70 miles per hour, and it takesa certain length of time, t, to travel. When traveling from the officebuilding to her home, ther is more traffic on the road, making Audratravel at a slower speed. It takes her 15 minutes more to drovefrom the office building to her house. Since Audra is always drivingthe same distance each way,.153

76 Part BDetermine the inverse functionto model the timerequired for Audra to travel from her house to the officebuilding. Show your work.B Click to revealanswersCAnswerAD154

77 Part AAnswerA train runs its route regularly between Philadelphia, PA andAtlantic City, NJ. When traveling from Philadelphia to Atlantic City,its average speed is 60 miles per hour, and it takes a certain lengthof time, t, to travel. When traveling from Atlantic City toPhiladelphia, it travels at a slower speed and takes 10 minutesmore to arrive at Philadelphia. Since the train is always travelingthe same distance each way,.Use the information provided to write the functionthat allowsyou to determine , the speed of the train during its trip fromAtlantic City to Philadelphia.ACBD155

78 Part BDetermine the inverse functionto model the timerequired to travel from Philadelphia to Atlantic City.Show your work.B Click to revealanswersCAnswerAD156

For a seesaw to be balanced,. Use the informationin the table to write the functionthat allows you todetermine, the mass of the first child.ACBDAnswer79 Part A: Two children sit on a seesaw, as illustrated. The mass, inkilograms, of the first child isand the mass, in kilograms, of thesecond child is. In the diagram,represent thedistance, in feet from the fulcrum (the balance point) to each child.The total distance between the children is 10 feet.From PARCC PBA Sample Test Calculator #5 Response Format157

80 Part BDetermine the inverse functionto model thedistance,, based on the mass of the first child. Showyour work.B Click to revealanswersAnswerACDFrom PARCC PBA Sample Test Calculator #5 Response Format158

function's inverse is also a function, just from looking at its graph. The Horizontal Line Test is used to determine if the inverse of a function is also considered to be a function. If a horizontal line crosses the function more than once, its inverse is NOT a