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One-to-One Functions

A function f is 1 -to- 1 if no two elements in the domain of f correspond to the same element in the range of f .  In other words, each x in the domain has exactly one image in the range.  And, no y in the range is the image of more than one x in the domain.

If the graph of a function f is known, it is easy to determine if the function is 1 -to- 1 .  Use the Horizontal Line Test.  If no horizontal line intersects the graph of the function f in more than one point, then the function is 1 -to- 1 .

Math diagram

Math diagram

A function f has an inverse f 1 (read f inverse) if and only if the function is 1 -to- 1 .

Properties of a 1 -to- 1 Function:

      1) The domain of f equals the range of f –1 and the range of f equals the domain of f 1 .

      2) 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 .

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

      Math diagram

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