Composite and Inverse FunctionsComposition of FunctionsSlide 3Slide 4Inverse FunctionsPerspectives for Input and OutputSlide 7Inverse Function NotationFinding Inverse Values from a TableFinding Inverse Values from a GraphFinding the Inverse FormulaDomain and Range of An Inverse FunctionAssignmentComposite and Inverse FunctionsLesson 2.42Composition of Functions•Consider two functions where the output of one is the input of the next•ExampleSquare yds/hr mowed is a function of how fast you push the mowerA = f(s)The time required to mow is a function of square yds/hr you cover T = g(A)speedsq yds/hrf(s)Timeg(A)3Composition of FunctionsGiven the following functions•Q = f(p) The number of barrels of oil sold when the price is p dollars per barrel•R(Q) is the revenue earned when Q barrels are sold•What is R(f(p)) ?What are the units of each function?4Composition of Functions•Given•Find the following compositions 2( ) 1 ( ) 2 3f x x g x x= + = +( (1))f g( (2))g f( ( ))g g nTry using your calculatorTry using your calculator5Inverse Functions•What if we cram a numberup the spout of a function and out of the funnel popsthe number that wouldhave given us the result??•The function that does this is called theinverse function1-34331Use spreadsheet to evaluate inverse of a functionUse spreadsheet to evaluate inverse of a function6Perspectives for Input and Output•Suppose you are told 1 gallon of paint covers 250 ft2You might derive the function•It is just as reasonable to consider how many gallons are needed for a certain area250( ) 250Area gf g g= �=250( )250areaGallonsAh A==7Perspectives for Input and Output•The mathematical relationship is the sameThe input on one f(g) is the output on h(A)•We would say the functions have an inverse relationship 250( ) 250Area gf g g= �=250( )250areaGallonsAh A==8Inverse Function Notation•For the inverse of function f, we use the notation f -1•Note that this is not the same as a negative exponentIt is not 1f1( ) 250( )250A f g gAg f A-= == =9Finding Inverse Values from a Table•Given the following table which defines the function f•Determinef(-2)f -1(2)f -1(-4)f(-1)x -2 -1 0 1 2f(x) 6 -4 3 2 910Finding Inverse Values from a Graph•Write some ordered pairsfor the functiondefined by thisgraph•Determinef -1(0)f -1(-2)x f(x)• Are there multiple answers • Is the inverse even a function?• Are there multiple answers • Is the inverse even a function?11Finding the Inverse Formula•Given the formula•Find the inverse function f -1(V)•StrategyWrite in formula notationSolve for the independent variabler = ?34( )3V f r rp= =343V rp=12Domain and Range of An Inverse Function•Note that the domain of the original function becomes the range of the inverseThus restrictions on the original domain affect the range of the inverse•AlsoThe range of the original may be restrictedThis affects the domain of the inverse•Consider the inverses of these functions5( )3xf xx+=-2( ) 25g x x= -As we saw on slide 10, some inverses might not even be functionsAs we saw on slide 10, some inverses might not even be functions13Assignment•Lesson 2.4•Page 90•Exercises 1 – 37
View Full Document
Unlocking...