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FunctionBijective
✖
FunctionBijective

✖

tests whether has exactly one solution x∈Reals for each y∈Reals.

✖

tests whether has exactly one solution x∈dom for each y∈dom.

✖
FunctionBijective[{f1,f2,…},{x1,x2,…},dom]

tests whether has exactly one solution x1,x2,…∈dom for each y1,y2,…∈dom.

✖
FunctionBijective[{funs,xcons,ycons},xvars,yvars,dom]

tests whether has exactly one solution with xvars∈dom restricted by the constraints xcons for each yvars∈dom restricted by the constraints ycons.

Details and Options

  • A bijective function is also known as one-to-one and onto.
  • A function is bijective if for each there is exactly one such that .
  • FunctionBijective[{funs,xcons,ycons},xvars,yvars,dom] returns True if the mapping is bijective, where is the solution set of xcons and is the solution set of ycons.
  • If funs contains parameters other than xvars, the result is typically a ConditionalExpression.
  • Possible values for dom are Reals and Complexes. If dom is Reals, then all variables, parameters, constants and function values are restricted to be real.
  • The domain of funs is restricted by the condition given by FunctionDomain.
  • xcons and ycons can contain equations, inequalities or logical combinations of these.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    GenerateConditions Truewhether to generate conditions on parameters
    PerformanceGoal $PerformanceGoalwhether to prioritize speed or quality
  • Possible settings for GenerateConditions include:
  • Automaticnongeneric conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal are "Speed" and "Quality".

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Test bijectivity of a univariate function over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-b2fcqx
Out[1]=1

Test bijectivity over the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-0dl46
Out[1]=1

Test bijectivity of a polynomial mapping over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dp3h13
Out[1]=1

Test bijectivity of a polynomial with symbolic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-kqrk3c
Out[1]=1

Scope  (10)Survey of the scope of standard use cases

Bijectivity over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gxhl11
Out[1]=1

Some values are attained more than once:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-d5ri0g
Out[2]=2

Bijectivity between subsets of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-glvby
Out[1]=1

For , each positive value is attained exactly once:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-h18c7
Out[2]=2

Bijectivity over the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gb8vpl
Out[1]=1

Some values are not attained:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-unv0t
Out[2]=2

The logarithm is bijective onto :

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-i6otm2
Out[3]=3

Bijectivity over a subset of complexes:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-e5opda
Out[1]=1

is not bijective over the whole complex plane:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-outewg
Out[2]=2

Some values are attained more than once:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gsupzm
Out[3]=3

Bijectivity over the integers:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-oypp7s
Out[1]=1

Bijectivity of linear mappings:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-fuu3lu
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dw1lhm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bb3g0a
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-huze44
Out[4]=4

A linear mapping is bijective iff its matrix is square and of the maximal rank:

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dwvhdm
Out[5]=5

Bijectivity of polynomial mappings :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dcji69
Out[1]=1

The restricted mapping TemplateBox[{}, NonNegativeReals]^2->TemplateBox[{}, Reals]^2\(TemplateBox[{}, PositiveReals]xTemplateBox[{}, NegativeReals]), equal to the real and imaginary part of , is bijective:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-63710
Out[2]=2

Bijectivity of polynomial mappings :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dcz40b
Out[1]=1

The Jacobian determinant of a bijective complex polynomial mapping must be constant:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-er3pcx
Out[2]=2

The Jacobian conjecture states that the reverse implication is true:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-w5rpi
Out[3]=3

Indeed, this polynomial mapping with a constant Jacobian is bijective:

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-jhg8kl
Out[4]=4

Bijectivity of a real polynomial with symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-dqtdtb
Out[1]=1

Bijectivity of a real polynomial mapping with symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-yju7
Out[1]=1

Options  (4)Common values & functionality for each option

Assumptions  (1)

FunctionBijective gives a conditional answer here:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bnv0jw
Out[1]=1

This checks the bijectivity for the remaining real values of :

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c50qtk
Out[2]=2

GenerateConditions  (2)

By default, FunctionBijective may generate conditions on symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-osy2z
Out[1]=1

With GenerateConditions->None, FunctionBijective fails instead of giving a conditional result:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-fe9ubz
Out[2]=2

This returns a conditionally valid result without stating the condition:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-na5ydu
Out[3]=3

By default, all conditions are reported:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-tdcquw
Out[1]=1

With GenerateConditions->Automatic, conditions that are generically true are not reported:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-291b1m
Out[2]=2

PerformanceGoal  (1)

Use PerformanceGoal to avoid potentially expensive computations:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-i86kxj
Out[1]=1

The default setting uses all available techniques to try to produce a result:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-i9gq
Out[2]=2

Applications  (11)Sample problems that can be solved with this function

Basic Applications  (8)

Check bijectivity of :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-eptf8l
Out[1]=1

Each value is attained exactly once:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gxd7sb
Out[2]=2

is not bijective:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-f83ch9
Out[3]=3

Some values, e.g. , are attained more than once:

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gxmk04
Out[4]=4

is not bijective:

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bwjk61
Out[5]=5

Some values, e.g. , are not attained:

In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-8ewe
Out[6]=6

is bijective in its real domain:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c1tdhr
Out[1]=1

Each value is attained exactly once:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c8mb9m
Out[2]=2

Show that TemplateBox[{x}, SinhIntegral] is bijective:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-0d38
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-m9agyl
Out[4]=4

Show that is not bijective:

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-rzkwi
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-xptsm
Out[6]=6

A function is bijective iff it is injective and surjective:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-eq3n1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gpoem8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cmj0cg
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gib1r2
Out[4]=4

is not bijective because it is not injective:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c1qdb3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bwxyy2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bprd2
Out[3]=3

is not bijective because it is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gz7ge
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-e34by
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bm67w1
Out[3]=3

A function is bijective if any horizontal line intersects the graph exactly once:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ocoy6
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-j9tn74
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ddq05o
Out[3]=3

TemplateBox[{x}, SinIntegral] is not bijective:

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cwxvsx
Out[4]=4

Some horizontal lines intersect the graph more than once; others do not intersect it at all:

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-l38ndl
Out[5]=5

Composition of bijective functions TemplateBox[{}, Reals]->TemplateBox[{}, Reals] is bijective:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-hp6wwn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-diggiz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-b2t25e
Out[3]=3

An affine mapping TemplateBox[{}, Reals]^n->TemplateBox[{}, Reals]^n given by is bijective if the rank of equals :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-m9q295
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-hq1fwk
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bet2ie
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-fclhyh
In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-d64ko
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cp7s2y
Out[6]=6

Probability  (1)

CDF of a distribution with a strictly positive PDF is bijective onto :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ite5ik
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-lw7z4y
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-h64nvi
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-l16efo
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-e654vt
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bybrxp
Out[6]=6

SurvivalFunction is bijective onto as well:

In[7]:=7
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-f5j5uf
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cjzze
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-jxsvmd
Out[9]=9

Quantile is bijective onto the reals:

In[10]:=10
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-o5v0zh
Out[10]=10
In[11]:=11
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bbrndf
Out[11]=11
In[12]:=12
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cmsvcg
Out[12]=12

Calculus  (2)

Compute by change of variables:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-do4pj

If is a bijective mapping , then int_Uf(g(u)) TemplateBox[{TemplateBox[{{{(, {partial, g}, )}, /, {(, {partial, u}, )}}}, Det]}, Abs]du=int_Vf(v)dv:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-hj81ga
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ev5xrl
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cxipq7
Out[4]=4

Check that is a bijective mapping TemplateBox[{}, Reals]^2->TemplateBox[{}, Reals]^2:

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cdekr3
Out[5]=5

Compute the integral:

In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bwpwfp
Out[6]=6

Compute the original integral directly:

In[7]:=7
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-f66t23
Out[7]=7

Compute the surface area of a ball with radius using a rational parametrization:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-9grqe
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-end738
Out[2]=2

Check that the parametrization is bijective onto the ball less a lower-dimensional set:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-db23hs
Out[3]=3

The surface area is equal to the integral of square root of Gram determinant of :

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-boflnz
In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ei12ro
Out[5]=5

Properties & Relations  (3)Properties of the function, and connections to other functions

is bijective iff the equation has exactly one solution for each :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bgha30
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-fgftkt
Out[2]=2

Use Solve to find the solutions:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-h9imdc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-ec0f5e
Out[4]=4

A real continuous function on an interval is bijective iff it is monotonic and the limits at endpoints are and :

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-b0d3so
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-mtrqj2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-daqrl2
Out[3]=3

Use FunctionMonotonicity to determine the monotonicity of a function:

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-fmt43q
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-if28q4
Out[5]=5

Use Limit to compute the limits:

In[6]:=6
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-d2r6j0
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bc9gqs
Out[7]=7

A complex polynomial mapping is bijective iff it has a polynomial inverse:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bfsvvm
In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cdw82t
Out[2]=2

Use Solve to find the polynomial inverse:

In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cqiub8
Out[3]=3

Verify that is a two-sided inverse of :

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-cpx0gn
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-lbnzh
Out[5]=5

Possible Issues  (2)Common pitfalls and unexpected behavior

FunctionBijective determines the real domain of functions using FunctionDomain:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c1rnh7
Out[1]=1

is bijective onto in the real domain reported by FunctionDomain:

In[2]:=2
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-gfcxum
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-e806up
Out[3]=3

is real valued and not bijective onto over the whole reals:

In[4]:=4
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-bhmdzf
Out[4]=4

All subexpressions of need to be real valued for a point to belong to the real domain of :

In[5]:=5
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-h3a0uv
Out[5]=5

FunctionBijective restricts the domain to the inverse image of the solution set of ycons:

In[1]:=1
✖
https://wolfram.com/xid/0rkgicew0xuv8ve-c7ycwd
Out[1]=1

Division by two is a bijection between the even integers (the inverse image of integers) and the integers.

Wolfram Research (2020), FunctionBijective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionBijective.html.
✖
Wolfram Research (2020), FunctionBijective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionBijective.html.

Text

Wolfram Research (2020), FunctionBijective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionBijective.html.

✖
Wolfram Research (2020), FunctionBijective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionBijective.html.

CMS

Wolfram Language. 2020. "FunctionBijective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionBijective.html.

✖
Wolfram Language. 2020. "FunctionBijective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionBijective.html.

APA

Wolfram Language. (2020). FunctionBijective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionBijective.html

✖
Wolfram Language. (2020). FunctionBijective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionBijective.html

BibTeX

@misc{reference.wolfram_2025_functionbijective, author="Wolfram Research", title="{FunctionBijective}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionBijective.html}", note=[Accessed: 22-May-2025 ]}

✖
@misc{reference.wolfram_2025_functionbijective, author="Wolfram Research", title="{FunctionBijective}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionBijective.html}", note=[Accessed: 22-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_functionbijective, organization={Wolfram Research}, title={FunctionBijective}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionBijective.html}, note=[Accessed: 22-May-2025 ]}

✖
@online{reference.wolfram_2025_functionbijective, organization={Wolfram Research}, title={FunctionBijective}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionBijective.html}, note=[Accessed: 22-May-2025 ]}