FunctionDomain
✖
FunctionDomain
finds the largest domain of definition of the real function f of the variable x.
finds the largest domain of definition of the mapping funs of the variables vars.
finds the domain of funs with the values of vars restricted by constraints cons.
Details and Options

- funs should be a list of functions of variables vars.
- 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
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (4)Survey of the scope of standard use cases

https://wolfram.com/xid/0b8cxiqdn6-basbu


https://wolfram.com/xid/0b8cxiqdn6-ccznwu


https://wolfram.com/xid/0b8cxiqdn6-b47lx8

Domain restricted by constraints:

https://wolfram.com/xid/0b8cxiqdn6-m34pw


https://wolfram.com/xid/0b8cxiqdn6-it5oj


https://wolfram.com/xid/0b8cxiqdn6-eryiah


https://wolfram.com/xid/0b8cxiqdn6-oa685j


https://wolfram.com/xid/0b8cxiqdn6-ootuff


https://wolfram.com/xid/0b8cxiqdn6-yj4n5

Complex multivariate functions:

https://wolfram.com/xid/0b8cxiqdn6-b9ty1t


https://wolfram.com/xid/0b8cxiqdn6-dtpxfh

Options (2)Common values & functionality for each option
GeneratedParameters (1)
FunctionDomain may introduce new parameters to represent the domain:

https://wolfram.com/xid/0b8cxiqdn6-k20zi0

Use GeneratedParameters to control how the parameters are generated:

https://wolfram.com/xid/0b8cxiqdn6-d8b1xx

Method (1)
By default, domains of real univariate functions are given in a reduced form:

https://wolfram.com/xid/0b8cxiqdn6-b9aba5

Domains of other functions are not reduced:

https://wolfram.com/xid/0b8cxiqdn6-ick1f

Use Method to specify whether the domain should be given in a reduced form:

https://wolfram.com/xid/0b8cxiqdn6-gbv0nf


https://wolfram.com/xid/0b8cxiqdn6-bptna9

Applications (13)Sample problems that can be solved with this function
Basic Applications (6)

https://wolfram.com/xid/0b8cxiqdn6-ikhlmu

The imaginary part of the function is zero on the real domain:

https://wolfram.com/xid/0b8cxiqdn6-62zki


https://wolfram.com/xid/0b8cxiqdn6-bj6l9o

The complement of the domain is the open disk of radius 3, centered at :

https://wolfram.com/xid/0b8cxiqdn6-p1gxa

Outside the real domain, a function may be complex, singular or undefined:

https://wolfram.com/xid/0b8cxiqdn6-d12cxd

Outside the real domain, the function is complex valued:

https://wolfram.com/xid/0b8cxiqdn6-hvgix1

Negative integers are not in the domain of :

https://wolfram.com/xid/0b8cxiqdn6-b4cct2

The function has pole singularities at negative integers:

https://wolfram.com/xid/0b8cxiqdn6-x5qau

is undefined outside its real domain:

https://wolfram.com/xid/0b8cxiqdn6-vk7jd


https://wolfram.com/xid/0b8cxiqdn6-mh2f93

Compute the complex domain of :

https://wolfram.com/xid/0b8cxiqdn6-hkoh3w

has pole singularities at points that do not belong to the domain:

https://wolfram.com/xid/0b8cxiqdn6-d7aq3w

Compute the complex domain of :

https://wolfram.com/xid/0b8cxiqdn6-eievnt

has an essential singularity at zero:

https://wolfram.com/xid/0b8cxiqdn6-b7y5mc

Compute the complex domain of :

https://wolfram.com/xid/0b8cxiqdn6-bjd1y

is undefined outside the domain:

https://wolfram.com/xid/0b8cxiqdn6-cfeed7

Solving Equations and Optimization (2)

https://wolfram.com/xid/0b8cxiqdn6-b54dpd
The solutions must belong to the real domain of :

https://wolfram.com/xid/0b8cxiqdn6-zuobm

The plot of shows that there is one solution:

https://wolfram.com/xid/0b8cxiqdn6-b66aw3

Solve automatically uses the domain information and finds the solution:

https://wolfram.com/xid/0b8cxiqdn6-j33kl9


https://wolfram.com/xid/0b8cxiqdn6-e3unqm

The minimum must belong to the real domain of :

https://wolfram.com/xid/0b8cxiqdn6-c6ua4

Find the roots of in the interior of the real domain:

https://wolfram.com/xid/0b8cxiqdn6-fjg809

Select the root at which the value of is minimal:

https://wolfram.com/xid/0b8cxiqdn6-dgwutr

Check that the value of at
is less than the values of
at the domain endpoints:

https://wolfram.com/xid/0b8cxiqdn6-lisu84


https://wolfram.com/xid/0b8cxiqdn6-nyw1ww

Minimize automatically uses the domain information and finds the minimum:

https://wolfram.com/xid/0b8cxiqdn6-k86q5

Calculus (5)
If the limit of a function over points from its real domain exists, it must be a real number or a real infinity:

https://wolfram.com/xid/0b8cxiqdn6-btpout


https://wolfram.com/xid/0b8cxiqdn6-es2nd0

Use Limit to verify that the limits of at
along real directions are real infinities:

https://wolfram.com/xid/0b8cxiqdn6-bwq3i5


https://wolfram.com/xid/0b8cxiqdn6-f8cc5e

If the integral over a subset of the real domain exists, it is a real number or a real infinity:

https://wolfram.com/xid/0b8cxiqdn6-gc08zj

https://wolfram.com/xid/0b8cxiqdn6-fsyhhg

Use Integrate to compute the integral of :

https://wolfram.com/xid/0b8cxiqdn6-cg7wiv

Verify that the integral is indeed a real number:

https://wolfram.com/xid/0b8cxiqdn6-b8m3ab

If the derivative of a function at a point in its real domain exists, it is real valued:

https://wolfram.com/xid/0b8cxiqdn6-fuzy8l

https://wolfram.com/xid/0b8cxiqdn6-hjcjw6


https://wolfram.com/xid/0b8cxiqdn6-g7uiwr

The derivative is indeed real valued over the domain of :

https://wolfram.com/xid/0b8cxiqdn6-pi7o18


https://wolfram.com/xid/0b8cxiqdn6-h0cilx

A function has to be defined and real valued in order to be real analytic:

https://wolfram.com/xid/0b8cxiqdn6-byjvqx

Over its real domain, is real analytic:

https://wolfram.com/xid/0b8cxiqdn6-ck4ihw

Check complex analyticity of :

https://wolfram.com/xid/0b8cxiqdn6-eqt83n

A function has to be defined to be complex analytic:

https://wolfram.com/xid/0b8cxiqdn6-bdpxl3

Over its domain, is complex analytic:

https://wolfram.com/xid/0b8cxiqdn6-zqvjy

Possible Issues (2)Common pitfalls and unexpected behavior
All subexpressions of need to be real-valued for a point to belong to the real domain of
:

https://wolfram.com/xid/0b8cxiqdn6-02sm8

Negative reals are not in the real domain of because
is not real valued:

https://wolfram.com/xid/0b8cxiqdn6-k7tkyp


https://wolfram.com/xid/0b8cxiqdn6-c59ut4

The real domain information for mathematical functions is accurate up to lower-dimensional sets:

https://wolfram.com/xid/0b8cxiqdn6-gfd21d

There is no full-dimensional subset of the space on which HankelH1 is real valued:

https://wolfram.com/xid/0b8cxiqdn6-bm2se2

Here is a 1-dimensional subset of the space on which HankelH1 is real valued:

https://wolfram.com/xid/0b8cxiqdn6-qgwrn

https://wolfram.com/xid/0b8cxiqdn6-df68w7

FunctionDomain is unable to detect that is real valued:

https://wolfram.com/xid/0b8cxiqdn6-fmo8v6

Wolfram Research (2014), FunctionDomain, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDomain.html.
Text
Wolfram Research (2014), FunctionDomain, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDomain.html.
Wolfram Research (2014), FunctionDomain, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDomain.html.
CMS
Wolfram Language. 2014. "FunctionDomain." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionDomain.html.
Wolfram Language. 2014. "FunctionDomain." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionDomain.html.
APA
Wolfram Language. (2014). FunctionDomain. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionDomain.html
Wolfram Language. (2014). FunctionDomain. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionDomain.html
BibTeX
@misc{reference.wolfram_2025_functiondomain, author="Wolfram Research", title="{FunctionDomain}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionDomain.html}", note=[Accessed: 08-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_functiondomain, organization={Wolfram Research}, title={FunctionDomain}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionDomain.html}, note=[Accessed: 08-July-2025
]}