Inverses & One-to-One

Inverse Functions ,[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],One-to-One Functions A function f is a one-to-one function if, for elements a and b from the domain of f , a  b implies f ( a )  f ( b ).

One-to-One Functions ,[object Object],[object Object],[object Object],[object Object],[object Object]

Horizontal Line Test ,[object Object],[object Object],[object Object],If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Not one-to-one One-to-one

Inverse Functions ,[object Object],[object Object],Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

Finding an Equation for the Inverse Function ,[object Object],Finding the Equation of the Inverse of y = f ( x ) For a one-to-one function f defined by an equation y = f ( x ) , find the defining equation of the inverse as follows. ( Any restrictions on x and y should be considered.) 1. Interchange x and y. 2. Solve for y. 3 . Replace y with f -1 ( x ).

Example of Finding f -1 ( x ) ,[object Object],[object Object],Write f ( x ) = y . Interchange x and y . Solve for y . Replace y with f -1 ( x ).

The Graph of f -1 ( x ) ,[object Object],[object Object],If a function is one-to-one, the graph of its inverse f -1 ( x ) is a reflection of the graph of f across the line y = x .

Finding the Inverse of a Function with a Restricted Domain ,[object Object],[object Object],[object Object],[object Object],[object Object]

Important Facts About Inverses ,[object Object],[object Object],[object Object]

Application of Inverse Functions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Inverses & One-to-One

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