Inverse of a function
From AoPSWiki
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 ![g(x) = \sqrt[3]{x-6}](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/c/8/4/c8447f996310f4d453b3e9f9257f647d9fdc575d.gif)
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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