Inverse Function
Inverse Function

Given a function , the inverse of is the function which has the property that exactly when (for the same and ). That is, the inverse of a function exactly undoes whatever the function does. The inverse of the function is commonly denoted by .
Note : the inverse of a function is not the same as the reciprocal of that function. For example, the inverse of is (for , not . The notation is intended to represent the concept of "inverting the action of ", not "inverting the result of ".

Algebraically Finding Inverses—Restricting the Domain


To find a formula for the inverse of you switch the roles of and and then try to solve for . For example, to find the inverse of the function you first switch and : . Then you solve for : , or rewriting in standard form, .
Very often it is not possible to carry out the step of solving for , as there may be more than one solution. This means that the original function is not invertible on its natural domain. In such cases, it is usually possible to restrict the domain of the original function to one on which the solve step can be carried to completion.
For example, suppose you want to find the inverse, if it exists, of the function . Interchanging and and trying to solve for leads you to , so you can conclude that the original function, is not invertible on the entire real line. However, if you restrict the domain to nonnegative numbers, so when , then the inverse will be . Note we could also restrict the domain to be nonpositive numbers, so the function is when , in which case the inverse is . This illustrates that there is invariably some arbitrariness in the choice of restricted domain, and that there can be more than one—sometimes infinitely many—definitions of an "inverse" for a function which is not invertible on its natural domain.





Finding a graph's inverse


Drag on the graph below to draw an invertible function.
Click Invert to show the process of the function being inverted.
Click Clear to clear the graph and draw again.


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