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A **many to one function** is where several members of the domain map to the same member of the range. Another way of saying this is that different inputs can give the same output.

A one to one function, where distinctness is preserved and every input is matched with a unique output, is called an injection. So *a many to one function is not injective.*

These mappings don’t have well defined inverses, because knowing the output of the function doesn’t always tell us what the input is.

## Examples of a Many to One Function

Periodic functions, which repeat at well-defined intervals, are always many-to-one. The trigonometric functions are examples of this; for example, take the function f(x) = sin x. You can prove it is many to one by noting that sin x = sin (2 π + x) = sin (4 π + x), etc., or by noting that when you graph the function, you can draw a straight horizontal line that intersects it many times.

In fact, any graphed mapping which a horizontal straight line intersects more than once is a many to one mapping. Since a function always satisfies the vertical line test (no vertical line can intersect more than once), a many to one function is one which any vertical line intersects no more than once but some horizontal line intersects more than once.

## References

Pilkington, Annette. Math 1056 Lecture Notes: Lecture 1. University of Notre Dame Math Department. Retrieved from https://www3.nd.edu/~apilking/Math10560/Lectures/1.%20Inverse%20Functions.pdf. on June 16, 2019

Smith, Ken. Introduction to Functions. Precalculus Lecture Notes. Sam Houston State University Math Department. Retrieved from https://www.shsu.edu/~kws006/Precalculus/1.1_Function_Definition_files/S%26Z%201.3-1.5.pdf on June 16, 2019

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