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FunctionRange
✖
FunctionRange

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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
    Method Automaticwhat method should be used
    WorkingPrecision Automaticprecision to be used in computations
  • With WorkingPrecision->Automatic, FunctionRange may use numerical optimization to estimate the range.

Examples

open allclose all

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

Find the range of a real function:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-4uoqj
Out[1]=1

The range of a complex function:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-inecd9
Out[1]=1

Scope  (7)Survey of the scope of standard use cases

Real univariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-basbu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-ccznwu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-b47lx8
Out[3]=3

Range estimated numerically:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-m34pw
Out[1]=1

Range over a domain restricted by conditions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-fodx21
Out[1]=1

Complex univariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-chteaq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-it5oj
Out[2]=2

Real multivariate functions:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-yj4n5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-ootuff
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-0kaqn
Out[3]=3

Real multivariate mappings:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-qd53ys
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-iyhebl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-eptkig
Out[3]=3

Range over a domain restricted by conditions:

In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-phft6
Out[4]=4

Complex multivariate functions and mappings:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-cls99g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-b77w4q
Out[2]=2

Options  (2)Common values & functionality for each option

Method  (1)

By default, the results returned by FunctionRange may not be reduced:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-flkpuy
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-khj27o
Out[2]=2

WorkingPrecision  (1)

By default, FunctionRange attempts to compute exact results:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-gbccne
Out[1]=1

With finite WorkingPrecision, slower symbolic methods are not used:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-hskaqr
Out[2]=2

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

Basic Applications  (7)

Find the range of a real function:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-g56lk
Out[1]=1

All real values within the range are attained:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-cwgjht
Out[2]=2

Find the range of a discontinuous function:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-g1cey6
Out[1]=1

The range consists of two intervals:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-2mtgb
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-bxiioh
Out[1]=1

Between and the plot is contained within the range:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-kfmml0
Out[2]=2

Find the range of a complex function:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-bjnhht
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-ialcsq
Out[2]=2

The function does not attain values and :

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-gvie3d
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-bsnzad
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-c1kcqh
Out[2]=2

The images are a disk and a half-plane:

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-kppskv
Out[3]=3

A function is surjective if FunctionRange gives True:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-d7q6of
Out[1]=1

You can test surjectivity using FunctionSurjective:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-gsf03g
Out[2]=2

A surjective function attains all values:

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-ger9ee
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-id0sv
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-en4nke
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-g5qeq2
Out[3]=3

Confirm that is surjective onto using FunctionSurjective:

In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-frnww
Out[4]=4

All values in are attained:

In[5]:=5
✖
https://wolfram.com/xid/0i1l6ubg6-ffsihw
Out[5]=5

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

In[6]:=6
✖
https://wolfram.com/xid/0i1l6ubg6-jpmq3t
Out[6]=6

The value is not attained:

In[7]:=7
✖
https://wolfram.com/xid/0i1l6ubg6-cqtn1y
Out[7]=7

Confirm that is not surjective onto using FunctionSurjective:

In[8]:=8
✖
https://wolfram.com/xid/0i1l6ubg6-g8ckcy
Out[8]=8

Solving Equations and Optimization  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-dyumvf
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-co8gwc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-hjitdw
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-dqpcw4
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0i1l6ubg6-e0216y
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0i1l6ubg6-grn1ck
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0i1l6ubg6-bkqvfg
Out[7]=7

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-ictrne
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-bop27j
Out[2]=2

belongs to the range of , and hence has solutions:

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-ocmlor
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-bec5dg
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0i1l6ubg6-iua9
Out[5]=5

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

In[6]:=6
✖
https://wolfram.com/xid/0i1l6ubg6-qyi525
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0i1l6ubg6-ktkrb
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0i1l6ubg6-btzo66
Out[8]=8

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-drqvmo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-kohpxx
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-dpzuxs
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-lvqrbx
Out[4]=4

Calculus  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-n3ctk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-ga1div
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-ii41l
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-hg3ntj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0i1l6ubg6-do3fu2
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0i1l6ubg6-55ob7
Out[6]=6

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

In[7]:=7
✖
https://wolfram.com/xid/0i1l6ubg6-dgr96n
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0i1l6ubg6-f7cbao
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0i1l6ubg6-d4mxby
Out[9]=9

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-f4lef7
Out[1]=1

The limit may not belong to the range itself:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-duwhq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-b55qan
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-bh5fq3
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-gbn3ur
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-bzl9vc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-j6ux8f
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0i1l6ubg6-g7hoov
Out[5]=5

Properties & Relations  (1)Properties of the function, and connections to other functions

A function is surjective if its FunctionRange is True:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-jwh4ee
Out[1]=1

Use FunctionSurjective to test whether a functions is surjective:

In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-e842wa
Out[2]=2

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:

In[1]:=1
✖
https://wolfram.com/xid/0i1l6ubg6-c9tymu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0i1l6ubg6-fcv23u
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0i1l6ubg6-c74yh1
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0i1l6ubg6-h8303p
Out[4]=4
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.

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 ]}

✖
@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 ]}

✖
@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 ]}