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.
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:
-
- With WorkingPrecision->Automatic , FunctionRange may use numerical optimization to estimate the range.
Examples
open all close allBasic 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:
Possible Issues (1)
Values at isolated points at which the function is real-valued may not be included in the result:
is non-real valued for , except for isolated values of :
Real values of for may lie outside the range given by FunctionRange :
Related Guides
History
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_2025_functionrange, author="Wolfram Research", title="{FunctionRange}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionRange.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_functionrange, organization={Wolfram Research}, title={FunctionRange}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionRange.html}, note=[Accessed: 05-December-2025]}