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# 1140 L12 - y=x y = f(x(a,b y=x

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Lecture 12: Section 1.9 Inverse Functions Def. If a function f is a set of ordered pairs ( x; y ), then inverse relation of f is the set of ordered pairs ( y; x ). ex. Find the inverse relation of the following func- tions. Is the inverse a function? 1) f : f ( 2 ; 2) ; (3 ; 1) ; (4 ; 0) ; ( 1 ; 2) g 2) g : f ( 2 ; 2) ; (3 ; 1) ; (0 ; 0) ; (4 ; 1) g NOTE: An inverse of a function may not be a function.

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One-to-One Functions Def. A function f is a one-to-one function if each value of the dependent variable ( y ) corresponds to exactly one value of the independent variable ( x ). That is, given any two elements x 1 and x 2 in the domain of f , if x 1 6 = x 2 , then f ( x 1 ) 6 = f ( x 2 ). NOTE: If a function f is a set of ordered pairs, f is one-to-one if no two ordered pairs have the same second element. An inverse of a function f is also a function if and only if f is a one-to-one function. ex. Let f ( x ) = x 2 . Is f a one-to-one function?
Horizontal Line Test A function f is one-to-one if and only if no horizontal line intersects the graph of f more than once.

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