FunctionRange
✖
FunctionRange
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:
-
GeneratedParameters C how to name parameters that are generated Method Automatic what method should be used WorkingPrecision Automatic precision to be used in computations - With WorkingPrecision->Automatic, FunctionRange may use numerical optimization to estimate the range.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (7)Survey of the scope of standard use cases

https://wolfram.com/xid/0i1l6ubg6-basbu


https://wolfram.com/xid/0i1l6ubg6-ccznwu


https://wolfram.com/xid/0i1l6ubg6-b47lx8


https://wolfram.com/xid/0i1l6ubg6-m34pw


Range over a domain restricted by conditions:

https://wolfram.com/xid/0i1l6ubg6-fodx21


https://wolfram.com/xid/0i1l6ubg6-chteaq


https://wolfram.com/xid/0i1l6ubg6-it5oj


https://wolfram.com/xid/0i1l6ubg6-yj4n5


https://wolfram.com/xid/0i1l6ubg6-ootuff


https://wolfram.com/xid/0i1l6ubg6-0kaqn



https://wolfram.com/xid/0i1l6ubg6-qd53ys


https://wolfram.com/xid/0i1l6ubg6-iyhebl


https://wolfram.com/xid/0i1l6ubg6-eptkig

Range over a domain restricted by conditions:

https://wolfram.com/xid/0i1l6ubg6-phft6

Complex multivariate functions and mappings:

https://wolfram.com/xid/0i1l6ubg6-cls99g


https://wolfram.com/xid/0i1l6ubg6-b77w4q

Options (2)Common values & functionality for each option
Method (1)
By default, the results returned by FunctionRange may not be reduced:

https://wolfram.com/xid/0i1l6ubg6-flkpuy

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

https://wolfram.com/xid/0i1l6ubg6-khj27o

WorkingPrecision (1)
By default, FunctionRange attempts to compute exact results:

https://wolfram.com/xid/0i1l6ubg6-gbccne

With finite WorkingPrecision, slower symbolic methods are not used:

https://wolfram.com/xid/0i1l6ubg6-hskaqr

Applications (13)Sample problems that can be solved with this function
Basic Applications (7)
Find the range of a real function:

https://wolfram.com/xid/0i1l6ubg6-g56lk

All real values within the range are attained:

https://wolfram.com/xid/0i1l6ubg6-cwgjht

Find the range of a discontinuous function:

https://wolfram.com/xid/0i1l6ubg6-g1cey6

The range consists of two intervals:

https://wolfram.com/xid/0i1l6ubg6-2mtgb

Find the range of over the interval
:

https://wolfram.com/xid/0i1l6ubg6-bxiioh

Between and
the plot is contained within the range:

https://wolfram.com/xid/0i1l6ubg6-kfmml0

Find the range of a complex function:

https://wolfram.com/xid/0i1l6ubg6-bjnhht

https://wolfram.com/xid/0i1l6ubg6-ialcsq

The function does not attain values and
:

https://wolfram.com/xid/0i1l6ubg6-gvie3d

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

https://wolfram.com/xid/0i1l6ubg6-bsnzad


https://wolfram.com/xid/0i1l6ubg6-c1kcqh

The images are a disk and a half-plane:

https://wolfram.com/xid/0i1l6ubg6-kppskv

A function is surjective if FunctionRange gives True:

https://wolfram.com/xid/0i1l6ubg6-d7q6of

You can test surjectivity using FunctionSurjective:

https://wolfram.com/xid/0i1l6ubg6-gsf03g

A surjective function attains all values:

https://wolfram.com/xid/0i1l6ubg6-ger9ee

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

https://wolfram.com/xid/0i1l6ubg6-id0sv

https://wolfram.com/xid/0i1l6ubg6-en4nke

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

https://wolfram.com/xid/0i1l6ubg6-g5qeq2

Confirm that is surjective onto
using FunctionSurjective:

https://wolfram.com/xid/0i1l6ubg6-frnww


https://wolfram.com/xid/0i1l6ubg6-ffsihw

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

https://wolfram.com/xid/0i1l6ubg6-jpmq3t


https://wolfram.com/xid/0i1l6ubg6-cqtn1y

Confirm that is not surjective onto
using FunctionSurjective:

https://wolfram.com/xid/0i1l6ubg6-g8ckcy

Solving Equations and Optimization (3)
The equation has solutions in the real domain of
if and only if
belongs to the real range of
:

https://wolfram.com/xid/0i1l6ubg6-dyumvf

belongs to the range of
, and hence
has solutions:

https://wolfram.com/xid/0i1l6ubg6-co8gwc


https://wolfram.com/xid/0i1l6ubg6-hjitdw


https://wolfram.com/xid/0i1l6ubg6-dqpcw4

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

https://wolfram.com/xid/0i1l6ubg6-e0216y


https://wolfram.com/xid/0i1l6ubg6-grn1ck


https://wolfram.com/xid/0i1l6ubg6-bkqvfg

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

https://wolfram.com/xid/0i1l6ubg6-ictrne

https://wolfram.com/xid/0i1l6ubg6-bop27j

belongs to the range of
, and hence
has solutions:

https://wolfram.com/xid/0i1l6ubg6-ocmlor


https://wolfram.com/xid/0i1l6ubg6-bec5dg


https://wolfram.com/xid/0i1l6ubg6-iua9

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

https://wolfram.com/xid/0i1l6ubg6-qyi525


https://wolfram.com/xid/0i1l6ubg6-ktkrb


https://wolfram.com/xid/0i1l6ubg6-btzo66

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

https://wolfram.com/xid/0i1l6ubg6-drqvmo


https://wolfram.com/xid/0i1l6ubg6-kohpxx

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

https://wolfram.com/xid/0i1l6ubg6-dpzuxs


https://wolfram.com/xid/0i1l6ubg6-lvqrbx

Calculus (3)
The range of a continuous function over a connected interval must be a connected interval:

https://wolfram.com/xid/0i1l6ubg6-n3ctk


https://wolfram.com/xid/0i1l6ubg6-ga1div


https://wolfram.com/xid/0i1l6ubg6-ii41l

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

https://wolfram.com/xid/0i1l6ubg6-hg3ntj


https://wolfram.com/xid/0i1l6ubg6-do3fu2


https://wolfram.com/xid/0i1l6ubg6-55ob7

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

https://wolfram.com/xid/0i1l6ubg6-dgr96n


https://wolfram.com/xid/0i1l6ubg6-f7cbao


https://wolfram.com/xid/0i1l6ubg6-d4mxby

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

https://wolfram.com/xid/0i1l6ubg6-f4lef7

The limit may not belong to the range itself:

https://wolfram.com/xid/0i1l6ubg6-duwhq


https://wolfram.com/xid/0i1l6ubg6-b55qan

Estimate the value of the integral of
over the interval
:

https://wolfram.com/xid/0i1l6ubg6-bh5fq3

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

https://wolfram.com/xid/0i1l6ubg6-gbn3ur

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

https://wolfram.com/xid/0i1l6ubg6-bzl9vc


https://wolfram.com/xid/0i1l6ubg6-j6ux8f

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

https://wolfram.com/xid/0i1l6ubg6-g7hoov

Properties & Relations (1)Properties of the function, and connections to other functions
A function is surjective if its FunctionRange is True:

https://wolfram.com/xid/0i1l6ubg6-jwh4ee

Use FunctionSurjective to test whether a functions is surjective:

https://wolfram.com/xid/0i1l6ubg6-e842wa

Possible Issues (1)Common pitfalls and unexpected behavior
Values at isolated points at which the function is real-valued may not be included in the result:

https://wolfram.com/xid/0i1l6ubg6-c9tymu


https://wolfram.com/xid/0i1l6ubg6-fcv23u

is non-real valued for
, except for isolated values of
:

https://wolfram.com/xid/0i1l6ubg6-c74yh1

Real values of for
may lie outside the range given by FunctionRange:

https://wolfram.com/xid/0i1l6ubg6-h8303p

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.
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.
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
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: 08-July-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: 08-July-2025
]}