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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