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

open allclose all

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:

Applications  (13)

Basic Applications  (7)

Find the range of a real function:

All real values within the range are attained:

Find the range of a discontinuous function:

The range consists of two intervals:

Find the range of TemplateBox[{x}, Fibonacci] over the interval :

Between and the plot is contained within the range:

Find the range of a complex function:

The function does not attain values and :

Compute the images of the unit disk through Möbius transformations and :

The images are a disk and a half-plane:

A function is surjective if FunctionRange gives True:

You can test surjectivity using FunctionSurjective:

A surjective function attains all values:

A function is surjective on a set of values if that set of values is contained in the function's range:

Use FindInstance to show that the interval is contained in the range of :

Confirm that is surjective onto using FunctionSurjective:

All values in are attained:

Use FindInstance to show that the interval is not contained in the range of :

The value is not attained:

Confirm that is not surjective onto using FunctionSurjective:

Solving Equations and Optimization  (3)

The equation has solutions in the real domain of if and only if belongs to the real range of :

belongs to the range of TemplateBox[{x}, LogGamma], and hence TemplateBox[{x}, LogGamma]=3 has solutions:

does not belong to the range of TemplateBox[{x}, LogGamma], and hence TemplateBox[{x}, LogGamma]=-1 has no solutions:

The equation has complex solutions if and only if belongs to the complex range of :

belongs to the range of , and hence has solutions:

does not belong to the range of , and hence has no solutions:

Compute the infimum and the supremum of values of a function:

You can also compute the infimum and the supremum of a function using MinValue and MaxValue:

Calculus  (3)

The range of a continuous function over a connected interval must be a connected interval:

The range of a discontinuous function over a connected interval may be disconnected:

The range of a discontinuous function over a connected interval may be connected too:

If a function has a limit, that limit must belong to the closure of the function's range:

The limit may not belong to the range itself:

Estimate the value of the integral of TemplateBox[{x}, SinIntegral] over the interval :

must be between the minimum and the maximum values in the range times the length of the interval:

Verify that the value of the integral computed using Integrate satisfies the inequalities:

is equal to the average value of the function in the interval times the length of the interval:

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.

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

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: 25-May-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_functionrange, organization={Wolfram Research}, title={FunctionRange}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionRange.html}, note=[Accessed: 25-May-2022 ]}