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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 f [ x ] is equal to y.
  • For a function with several arguments, InverseFunction[ f ] represents the inverse with respect to the first argument. InverseFunction[ f , n ] represents the inverse with respect to the n argument. InverseFunction[ f , n , tot ] represents the inverse with respect to the n argument when there are tot arguments in all.
  • In OutputForm and StandardForm, InverseFunction[ f ] is printed as .
  • As discussed in Section 3.2.7, many mathematical functions do not have unique inverses. In such cases, InverseFunction[ f ] can represent only one of the possible inverses for f.
  • Example: InverseFunction[Sin].
  • InverseFunction is generated by Solve when the option InverseFunctions is set to Automatic or True.
  • See the Mathematica book: Section 2.2.1, Section 2.2.9Section 3.4.5.
  • See also: Solve, InverseSeries, Composition, Derivative.

    Further Examples

    Here are two inverse functions evaluated at y.

    In[1]:=

    Out[1]=

    In[2]:=

    Out[2]=

    Composing a function with its inverse in either order is the identity operation.

    In[3]:=

    InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses.

    Out[3]=

    In[4]:=

    InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses.

    Out[4]=

    The inverse of a composition is another composition.

    In[5]:=

    Out[5]=