# InverseFunction

represents the inverse of the function f, defined so that InverseFunction[f][y] gives the value of x for which f[x] is equal to y.

InverseFunction[f,n,tot]

represents the inverse with respect to the n argument when there are tot arguments in all.

# Details # Examples

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## Basic Examples(3)

The "inverse function" of Sin is ArcSin:

Inverse of a pure function:

Symbolic inverse function:

Derivative of an inverse function:

## Scope(8)

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: ## Properties & Relations(3)

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:  ## Possible Issues(1)

Equations and may not hold for arbitrary and : Introduced in 1991
(2.0)