FunctionRange

FunctionRange[f,x,y]

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

FunctionRange[f,x,y,dom]

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

FunctionRange[funs,xvars,yvars,dom]

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

FunctionRange[{funs,cons},xvars,yvars,dom]

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.

Examples

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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:

Properties & Relations  (1)

A function is surjective if its FunctionRange is True:

Use FunctionSurjective to test whether a functions is surjective:

Wolfram Research (2014), FunctionRange, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionRange.html.

Text

Wolfram Research (2014), FunctionRange, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionRange.html.

BibTeX

@misc{reference.wolfram_2021_functionrange, author="Wolfram Research", title="{FunctionRange}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionRange.html}", note=[Accessed: 18-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_functionrange, organization={Wolfram Research}, title={FunctionRange}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionRange.html}, note=[Accessed: 18-September-2021 ]}

CMS

Wolfram Language. 2014. "FunctionRange." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionRange.html.

APA

Wolfram Language. (2014). FunctionRange. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionRange.html