# Inverse of a function

The **inverse of a function** is a function that "undoes" the action of a given function.

For example, consider the function given by the rule . The function has the property that . In this case, is called the **(right) inverse function** of . Similarly, a function such that is called the **left inverse function** of . Typically, the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function *the inverse function*. For example, in our example above, is both a right and left inverse to on the real numbers.

Often the inverse of a function is denoted by . Note that the does *not* indicate an exponent.

The inverse of a function with range is a function if and only if is injective, so that every element in the range is mapped from a distinct element in the domain. When the domain and range are subsets of the real numbers, one way to test this is the horizontal line test: if every horizontal line drawn in the plane intersects the graph of the function in at most one point, the function is injective.

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