6.1 One-to-One Functions; Inverse Function

A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

x1 y1 x1 y1 x2 y2 x2 x3 x3 y3 y3 One-to-one function NOT One-to-one Domain Range Domain Range One-to-one function NOT One-to-one function x1 y1 y2 x3 y3 Not a function Domain Range

Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

Use the graph to determine whether the function is one-to-one. Not one-to-one.

Use the graph to determine whether the function is one-to-one.

Let f denote a one-to-one function y = f(x) Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1. .

Domain of f Range of f

Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

y = x (0, 2) (2, 0)

The function is one-to-one. Find the inverse of The function is one-to-one. Interchange variables. Solve for y.

Check.