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. Often the inverse of a function
is denoted by
(the
does not indicate a exponent).
The inverse of a function is only a function itself iff the original function is one-to-one, or that every element in the range is paired with a distinct element in the domain. A way to test this is the horizontal line test, where if a horizontal line can be drawn through the graph of a function and touch two points on the graph, the function is not one-to-one.
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