Discrete Structure lecture_15

Apr 28th, 2015
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It includes all the details about the Discrete structure with examples.Step by step solution.

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DISCREAT STRUCTURELECTURE # 15INVERSE OFFUNCTIONKHAWAJA TARIQ MAHMOODLecture # 15INVERSE OF FUNCTIONSINVERSE FUNCTION:Suppose f : X Y is a bijective function. Then the inverse function f 1 : Y X is defined as: y Y , f 1 y x y f x That is, f 1 sends each element of Y back to the element of Xthat it came from under f .fx=f-1(y)f(x)=yf-1X=domain of fY=co-domain of fINVERSE FUNCTION FROM AN ARROW DIAGRAM:Let the bijection f : X Y be defined by the arrow diagram.fKhawaja Tariq Mahmood.1..62.7384.9XYPage 2Lecture # 15INVERSE OF FUNCTIONSThe inverse function f 1 : Y X is represented below by the arrow diagram.f- FUNCTION FROM A FORMULA:Let f : R R be defined by the formula f x 4 x 1 x R . Then f is bijective, therefore f 1exists. By definition of f 1 , f 1 y x f x yNow solvingf x yfor x 4x 1 y(by definition of f) 4x y 1 xHence, f 1 y y 14y 1is the inverse of f x 4 x 1 which defines f 1 : R R4Khawaja Tariq Mahmood.Page 3Lecture # 15INVERSE OF FUNCTIONSg-1g1..a2..b3..cXY.a1..b2..c3.YXINVERSEFUNCTIONRemark: Inverse of a function may not be a function.f-1f1.2.3.X.a.a.b.b.cd1.2..c3.d

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