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4_1 - Section 4.1 One-to-One Functions Inverse Functions...

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Section 4.1: One-to-One Functions; Inverse Functions Instructor: Ms. Hoa Nguyen ([email protected]) 1 One-to-One function Function : A function f can be thought of as the machine whose input is a number x and output is a UNIQUE number y = f ( x ), i.e., with each x , there is one and only one y = f ( x ). One-to-One Function : A function f is one-to-one if for 2 DIFFERENT inputs x 1 and x 2 , the outputs f ( x 1 ) and f ( x 2 ) are also DIFFERENT. Mathematically, if x 1 = x 2 then f ( x 1 ) = f ( x 2 ), i.e., if f ( x 1 ) = f ( x 2 ) then x 1 = x 2 . The graph of f ( x ) = x 3 : a one-to-one function. The graph of f ( x ) = x 2 : NOT a one-to-one function. 1
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Horizontal Line Test : A function f is one-to-one if and only if every horizontal line intersects the graph of f in AT MOST ONE point. A function f is increasing if f ( a ) < f ( b ) whenever a < b and a, b are in the domain of f . A function f is decreasing if f ( a ) > f ( b ) whenever a < b and a, b are in the domain of f . An increasing (decreasing) function is a one-to-one function.
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