The "inverse function" of Sin is ArcSin:

Inverse of a pure function:

Symbolic inverse function:

Derivative of an inverse function:

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 :

For non-algebraic input, Solve may use InverseFunction to represent solutions:

Equations and may not hold for arbitrary and :