finds the range of the real function f of the variable x returning the result in terms of y.


considers f to be a function with arguments and values in the domain dom.


finds the range of the mapping funs of the variables xvars returning the result in terms of yvars.


finds the range of the mapping funs with the values of xvars restricted by constraints cons.

Details and Options

  • funs should be a list of functions of variables xvars.
  • funs and yvars must be lists of equal lengths.
  • Possible values for dom are Reals and Complexes. The default is Reals.
  • If dom is Reals then all variables, parameters, constants, and function values are restricted to be real.
  • cons can contain equations, inequalities, or logical combinations of these.
  • The following options can be given:
  • GeneratedParametersChow to name parameters that are generated
    MethodAutomaticwhat method should be used
    WorkingPrecisionAutomaticprecision to be used in computations
  • With WorkingPrecision->Automatic, FunctionRange may use numerical optimization to estimate the range.


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Basic Examples  (2)

Find the range of a real function:

The range of a complex function:

Scope  (7)

Real univariate functions:

Range estimated numerically:

Range over a domain restricted by conditions:

Complex univariate functions:

Real multivariate functions:

Real multivariate mappings:

Range over a domain restricted by conditions:

Complex multivariate functions and mappings:

Options  (2)

Method  (1)

By default, the results returned by FunctionRange may not be reduced:

Use Method to specify that the result should be given in a reduced form:

WorkingPrecision  (1)

By default, FunctionRange attempts to compute exact results:

With finite WorkingPrecision, slower symbolic methods are not used:

Introduced in 2014