Inverse FunctionsDefinitionExampleSlide 4TerminologyDoes This Have An Inverse?Finding the InverseComposition of Inverse FunctionsGraphs of InversesSlide 10Investigating Inverse FunctionsDomain and RangeSlide 13Inverse PumpkinsSlide 15AssignmentInverse FunctionsLesson 10.2Definition•A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }•But ... what if we reverse the order of the pairs?This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }Example•Consider an element of an electrical circuit which increases its resistance as a function of temperature. T = Temp R = Resistance-20 500 15020 25040 350R = f(T)Example•We could also take the view that we wish to determine T, temperature as a function of R, resistance.R = Resistance T = Temp50 -20150 0250 20350 40T = g(R)Now we would say that g(R) and f(T) are inverse functions Now we would say that g(R) and f(T) are inverse functionsTerminology•If R = f(T) ... resistance is a function of temperature,•Then T = f-1(R) ... temperature is the inverse function of resistance.f-1(R) is read "f-inverse of R“is not an exponent it does not mean reciprocal 11xx-=Does This Have An Inverse?•Given the function at the rightCan it have an inverse?Why or Why Not?x Y1 52 94 67 5NO … when we reverse the ordered pairs, the result is Not a function.Finding the InverseTry22xyx+=-( )1Given ( ) 2 7then 2 77solve for x x272f x xy xyyf y-=- -=- -- -=- -=Composition of Inverse Functions•Consider •f(3) = 27 and f -1(27) = 3 •Thus, f(f -1(27)) = 27 •and f -1(f(3)) = 3 •In general f(f -1(n)) = n and f -1(f(n)) = n(assuming both f and f -1 are defined for n)3 13( ) ( )f x x and f x x-= =Graphs of Inverses•Again, consider•Set your calculator for the functions shown3 13( ) ( )f x x and f x x-= =Dotted style• Use Standard Zoom•Then use Square ZoomGraphs of Inverses•Note the two graphs are symmetric about the line y = xInvestigating Inverse Functions•Consider•Demonstrate that these are inverse functions •What happens with f(g(x))? •What happens with g(f(x))?33( ) 2 4( ) 48f x xxg x= -= +Define these functions on your calculator and try them outDefine these functions on your calculator and try them outDomain and Range•The domain of f is the range of f -1 •The range of f is the domain of f -1 •Thus ... we may be required to restrict the domain of f so that f -1 is a functionDomain and Range•Consider the function h(x) = x2 - 9• Determine the inverse function•Problem => f -1(x) is not a functionInverse Pumpkins•In a recent pumpkin launching contest, one launcher misfired so that the pumpkin went straight up into the air (!!) and came back down to land on the launch personnel! •Below is the graph of the height of the pumpkin as a function of time h(t) Note ... the curve is not the path of the pumpkin.Inverse Pumpkins•What is the "hang time" of the launch? •Restrict the domain of h(t) so that it has an inverse that is a function •Graph the inverse of this function •Change the story to go with your new graph. Explain in your story why it makes sense that the inverse is a function.Assignment•Lesson 10.2•Page 411•Exercises 1 – 49
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