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InverseFunction

InverseFunction[f]
represents the inverse of the function f, defined so that InverseFunction[f][y] gives the value of x for which is equal to y.
InverseFunction
represents the inverse with respect to the ^(th) argument when there are tot arguments in all.
The "inverse function" of Sin is ArcSin:
Inverse of a pure function:
Symbolic inverse function:
Derivative of an inverse function:
The "inverse function" of Sin is ArcSin:
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Inverse of a pure function:
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Symbolic inverse function:
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Derivative of an inverse function:
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Inverse of a one-to-one function:
When the function is not one-to-one InverseFunction issues a message:
For functions with a named principal branch of the inverse, the message is not issued:
Inverse function with respect to the second argument:
Inverse of a function with a restricted domain:
The domain of the inverse function is computed automatically:
Here a closed-form representation for the inverse function does not exist:
Evaluation of the inverse function at exact points yields exact numeric values:
However, the inverse may not be unique:
InverseFunction with respect to the first argument of a two-argument function:
Here a closed-form representation for the inverse function does not exist:
Evaluation at an exact point does not find an exact numeric representation:
Evaluation at an approximate point yields a numeric result:
Automatic simplification of symbolic inverses:
For arbitrary function and point , :
Note that neither nor for arbitrary and :
If solutions of exist, gives a solution of :
Use Reduce to find all solutions of :
Use FindInstance to find a solution of :
When solutions of do not exist, is not a solution of :
FindInstance returns an empty solution list:
For non-algebraic input, Solve may use InverseFunction to represent solutions:
Equations and may not hold for arbitrary and :
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