WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

FunctionDomain
✖
FunctionDomain

✖

finds the largest domain of definition of the real function f of the variable x.

✖
FunctionDomain[f,x,dom]

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

✖
FunctionDomain[funs,vars,dom]

finds the largest domain of definition of the mapping funs of the variables vars.

✖
FunctionDomain[{funs,cons},vars,dom]

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 Chow to name parameters that are generated
    Method Automaticwhat method should be used
    WorkingPrecisionAutomaticprecision to be used in computations

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the largest domain of definition of a real function:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-gbcipx
Out[1]=1

The largest domain of definition of a complex function:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-g2b4w7
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

Real univariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-basbu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-ccznwu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-b47lx8
Out[3]=3

Domain restricted by constraints:

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-m34pw
Out[4]=4

Complex univariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-it5oj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-eryiah
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-oa685j
Out[3]=3

Real multivariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-ootuff
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-yj4n5
Out[2]=2

Complex multivariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-b9ty1t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-dtpxfh
Out[2]=2

Options  (2)Common values & functionality for each option

GeneratedParameters  (1)

FunctionDomain may introduce new parameters to represent the domain:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-k20zi0
Out[1]=1

Use GeneratedParameters to control how the parameters are generated:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-d8b1xx
Out[2]=2

Method  (1)

By default, domains of real univariate functions are given in a reduced form:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-b9aba5
Out[1]=1

Domains of other functions are not reduced:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-ick1f
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-gbv0nf
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-bptna9
Out[4]=4

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

Basic Applications  (6)

Compute the real domain of :

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-ikhlmu
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-62zki
Out[3]=3

Compute the real domain of :

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-bj6l9o
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-p1gxa
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-d12cxd
Out[1]=1

Outside the real domain, the function is complex valued:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-hvgix1
Out[2]=2

Negative integers are not in the domain of TemplateBox[{{x, -, 1}}, AlternatingFactorial]:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-b4cct2
Out[3]=3

The function has pole singularities at negative integers:

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-x5qau
Out[4]=4

TemplateBox[{3, 2, x}, EllipticTheta] is undefined outside its real domain:

In[5]:=5
✖
https://wolfram.com/xid/0b8cxiqdn6-vk7jd
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0b8cxiqdn6-mh2f93
Out[6]=6

Compute the complex domain of :

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-hkoh3w
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-d7aq3w
Out[2]=2

Compute the complex domain of :

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-eievnt
Out[1]=1

has an essential singularity at zero:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-b7y5mc
Out[2]=2

Compute the complex domain of TemplateBox[{1, 2, x}, EllipticTheta]:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-bjd1y
Out[1]=1

TemplateBox[{1, 2, x}, EllipticTheta] is undefined outside the domain:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-cfeed7
Out[2]=2

Solving Equations and Optimization  (2)

Solve over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-b54dpd

The solutions must belong to the real domain of :

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-zuobm
Out[2]=2

The plot of shows that there is one solution:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-b66aw3
Out[3]=3

Solve automatically uses the domain information and finds the solution:

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-j33kl9
Out[4]=4

Find the global minimum of TemplateBox[{6, {-, 1}, x}, LegendreP3]:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-e3unqm
Out[1]=1

The minimum must belong to the real domain of :

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-c6ua4
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-fjg809
Out[3]=3

Select the root at which the value of is minimal:

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-dgwutr
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0b8cxiqdn6-lisu84
Out[5]=5

Visualize the result:

In[6]:=6
✖
https://wolfram.com/xid/0b8cxiqdn6-nyw1ww
Out[6]=6

Minimize automatically uses the domain information and finds the minimum:

In[7]:=7
✖
https://wolfram.com/xid/0b8cxiqdn6-k86q5
Out[7]=7

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:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-btpout
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-es2nd0
Out[2]=2

Use Limit to verify that the limits of TemplateBox[{x}, Gamma] at along real directions are real infinities:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-bwq3i5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-f8cc5e
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-gc08zj
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-fsyhhg
Out[2]=2

Use Integrate to compute the integral of :

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-cg7wiv
Out[3]=3

Verify that the integral is indeed a real number:

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-b8m3ab
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-fuzy8l
In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-hjcjw6
Out[2]=2

Compute the derivative:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-g7uiwr
Out[3]=3

The derivative is indeed real valued over the domain of :

In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-pi7o18
Out[4]=4

Check real analyticity of TemplateBox[{x, y}, BesselK]:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-h0cilx
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-byjvqx
Out[2]=2

Over its real domain, TemplateBox[{x, y}, BesselK] is real analytic:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-ck4ihw
Out[3]=3

Check complex analyticity of TemplateBox[{x, 4, 4, 1}, Hypergeometric2F1]:

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-eqt83n
Out[1]=1

A function has to be defined to be complex analytic:

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-bdpxl3
Out[2]=2

Over its domain, TemplateBox[{x, 4, 4, 1}, Hypergeometric2F1] is complex analytic:

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-zqvjy
Out[3]=3

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 :

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-02sm8
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-k7tkyp
Out[2]=2

is real valued for all real :

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-c59ut4
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b8cxiqdn6-gfd21d
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0b8cxiqdn6-bm2se2
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0b8cxiqdn6-qgwrn
In[4]:=4
✖
https://wolfram.com/xid/0b8cxiqdn6-df68w7
Out[4]=4

FunctionDomain is unable to detect that is real valued:

In[5]:=5
✖
https://wolfram.com/xid/0b8cxiqdn6-fmo8v6
Out[5]=5
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.

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 ]}

✖
@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 ]}

✖
@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 ]}