FunctionBijective
✖
FunctionBijective
tests whether has exactly one solution x1,x2,…∈dom for each y1,y2,…∈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 $Assumptions assumptions on parameters GenerateConditions True whether to generate conditions on parameters PerformanceGoal $PerformanceGoal whether to prioritize speed or quality - Possible settings for GenerateConditions include:
-
Automatic nongeneric conditions only True all conditions False no conditions None return unevaluated if conditions are needed - Possible settings for PerformanceGoal are "Speed" and "Quality".

Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Test bijectivity of a univariate function over the reals:

https://wolfram.com/xid/0rkgicew0xuv8ve-b2fcqx

Test bijectivity over the complexes:

https://wolfram.com/xid/0rkgicew0xuv8ve-0dl46

Test bijectivity of a polynomial mapping over the reals:

https://wolfram.com/xid/0rkgicew0xuv8ve-dp3h13

Test bijectivity of a polynomial with symbolic coefficients:

https://wolfram.com/xid/0rkgicew0xuv8ve-kqrk3c

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

https://wolfram.com/xid/0rkgicew0xuv8ve-gxhl11

Some values are attained more than once:

https://wolfram.com/xid/0rkgicew0xuv8ve-d5ri0g

Bijectivity between subsets of the reals:

https://wolfram.com/xid/0rkgicew0xuv8ve-glvby

For , each positive value is attained exactly once:

https://wolfram.com/xid/0rkgicew0xuv8ve-h18c7

Bijectivity over the complexes:

https://wolfram.com/xid/0rkgicew0xuv8ve-gb8vpl


https://wolfram.com/xid/0rkgicew0xuv8ve-unv0t

The logarithm is bijective onto :

https://wolfram.com/xid/0rkgicew0xuv8ve-i6otm2

Bijectivity over a subset of complexes:

https://wolfram.com/xid/0rkgicew0xuv8ve-e5opda

is not bijective over the whole complex plane:

https://wolfram.com/xid/0rkgicew0xuv8ve-outewg

Some values are attained more than once:

https://wolfram.com/xid/0rkgicew0xuv8ve-gsupzm

Bijectivity over the integers:

https://wolfram.com/xid/0rkgicew0xuv8ve-oypp7s

Bijectivity of linear mappings:

https://wolfram.com/xid/0rkgicew0xuv8ve-fuu3lu

https://wolfram.com/xid/0rkgicew0xuv8ve-dw1lhm


https://wolfram.com/xid/0rkgicew0xuv8ve-bb3g0a

https://wolfram.com/xid/0rkgicew0xuv8ve-huze44

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

https://wolfram.com/xid/0rkgicew0xuv8ve-dwvhdm

Bijectivity of polynomial mappings :

https://wolfram.com/xid/0rkgicew0xuv8ve-dcji69

The restricted mapping , equal to the real and imaginary part of
, is bijective:

https://wolfram.com/xid/0rkgicew0xuv8ve-63710

Bijectivity of polynomial mappings :

https://wolfram.com/xid/0rkgicew0xuv8ve-dcz40b

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

https://wolfram.com/xid/0rkgicew0xuv8ve-er3pcx

The Jacobian conjecture states that the reverse implication is true:

https://wolfram.com/xid/0rkgicew0xuv8ve-w5rpi

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

https://wolfram.com/xid/0rkgicew0xuv8ve-jhg8kl

Bijectivity of a real polynomial with symbolic parameters:

https://wolfram.com/xid/0rkgicew0xuv8ve-dqtdtb

Bijectivity of a real polynomial mapping with symbolic parameters:

https://wolfram.com/xid/0rkgicew0xuv8ve-yju7

Options (4)Common values & functionality for each option
Assumptions (1)
FunctionBijective gives a conditional answer here:

https://wolfram.com/xid/0rkgicew0xuv8ve-bnv0jw

This checks the bijectivity for the remaining real values of :

https://wolfram.com/xid/0rkgicew0xuv8ve-c50qtk

GenerateConditions (2)
By default, FunctionBijective may generate conditions on symbolic parameters:

https://wolfram.com/xid/0rkgicew0xuv8ve-osy2z

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

https://wolfram.com/xid/0rkgicew0xuv8ve-fe9ubz

This returns a conditionally valid result without stating the condition:

https://wolfram.com/xid/0rkgicew0xuv8ve-na5ydu

By default, all conditions are reported:

https://wolfram.com/xid/0rkgicew0xuv8ve-tdcquw

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

https://wolfram.com/xid/0rkgicew0xuv8ve-291b1m

PerformanceGoal (1)
Use PerformanceGoal to avoid potentially expensive computations:

https://wolfram.com/xid/0rkgicew0xuv8ve-i86kxj

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

https://wolfram.com/xid/0rkgicew0xuv8ve-i9gq

Applications (11)Sample problems that can be solved with this function
Basic Applications (8)

https://wolfram.com/xid/0rkgicew0xuv8ve-eptf8l

Each value is attained exactly once:

https://wolfram.com/xid/0rkgicew0xuv8ve-gxd7sb


https://wolfram.com/xid/0rkgicew0xuv8ve-f83ch9

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

https://wolfram.com/xid/0rkgicew0xuv8ve-gxmk04


https://wolfram.com/xid/0rkgicew0xuv8ve-bwjk61

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

https://wolfram.com/xid/0rkgicew0xuv8ve-8ewe

is bijective in its real domain:

https://wolfram.com/xid/0rkgicew0xuv8ve-c1tdhr

Each value is attained exactly once:

https://wolfram.com/xid/0rkgicew0xuv8ve-c8mb9m


https://wolfram.com/xid/0rkgicew0xuv8ve-0d38


https://wolfram.com/xid/0rkgicew0xuv8ve-m9agyl


https://wolfram.com/xid/0rkgicew0xuv8ve-rzkwi


https://wolfram.com/xid/0rkgicew0xuv8ve-xptsm

A function is bijective iff it is injective and surjective:

https://wolfram.com/xid/0rkgicew0xuv8ve-eq3n1


https://wolfram.com/xid/0rkgicew0xuv8ve-gpoem8


https://wolfram.com/xid/0rkgicew0xuv8ve-cmj0cg


https://wolfram.com/xid/0rkgicew0xuv8ve-gib1r2

is not bijective because it is not injective:

https://wolfram.com/xid/0rkgicew0xuv8ve-c1qdb3


https://wolfram.com/xid/0rkgicew0xuv8ve-bwxyy2


https://wolfram.com/xid/0rkgicew0xuv8ve-bprd2

is not bijective because it is not surjective:

https://wolfram.com/xid/0rkgicew0xuv8ve-gz7ge


https://wolfram.com/xid/0rkgicew0xuv8ve-e34by


https://wolfram.com/xid/0rkgicew0xuv8ve-bm67w1

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

https://wolfram.com/xid/0rkgicew0xuv8ve-ocoy6

https://wolfram.com/xid/0rkgicew0xuv8ve-j9tn74


https://wolfram.com/xid/0rkgicew0xuv8ve-ddq05o


https://wolfram.com/xid/0rkgicew0xuv8ve-cwxvsx

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

https://wolfram.com/xid/0rkgicew0xuv8ve-l38ndl

Composition of bijective functions is bijective:

https://wolfram.com/xid/0rkgicew0xuv8ve-hp6wwn


https://wolfram.com/xid/0rkgicew0xuv8ve-diggiz


https://wolfram.com/xid/0rkgicew0xuv8ve-b2t25e

An affine mapping given by
is bijective if the rank of
equals
:

https://wolfram.com/xid/0rkgicew0xuv8ve-m9q295

https://wolfram.com/xid/0rkgicew0xuv8ve-hq1fwk


https://wolfram.com/xid/0rkgicew0xuv8ve-bet2ie


https://wolfram.com/xid/0rkgicew0xuv8ve-fclhyh

https://wolfram.com/xid/0rkgicew0xuv8ve-d64ko


https://wolfram.com/xid/0rkgicew0xuv8ve-cp7s2y

Probability (1)
CDF of a distribution with a strictly positive PDF is bijective onto :

https://wolfram.com/xid/0rkgicew0xuv8ve-ite5ik


https://wolfram.com/xid/0rkgicew0xuv8ve-lw7z4y


https://wolfram.com/xid/0rkgicew0xuv8ve-h64nvi


https://wolfram.com/xid/0rkgicew0xuv8ve-l16efo


https://wolfram.com/xid/0rkgicew0xuv8ve-e654vt


https://wolfram.com/xid/0rkgicew0xuv8ve-bybrxp

SurvivalFunction is bijective onto as well:

https://wolfram.com/xid/0rkgicew0xuv8ve-f5j5uf


https://wolfram.com/xid/0rkgicew0xuv8ve-cjzze


https://wolfram.com/xid/0rkgicew0xuv8ve-jxsvmd

Quantile is bijective onto the reals:

https://wolfram.com/xid/0rkgicew0xuv8ve-o5v0zh


https://wolfram.com/xid/0rkgicew0xuv8ve-bbrndf


https://wolfram.com/xid/0rkgicew0xuv8ve-cmsvcg

Calculus (2)
Compute by change of variables:

https://wolfram.com/xid/0rkgicew0xuv8ve-do4pj
If is a bijective mapping
, then
:

https://wolfram.com/xid/0rkgicew0xuv8ve-hj81ga

https://wolfram.com/xid/0rkgicew0xuv8ve-ev5xrl

https://wolfram.com/xid/0rkgicew0xuv8ve-cxipq7

Check that is a bijective mapping
:

https://wolfram.com/xid/0rkgicew0xuv8ve-cdekr3


https://wolfram.com/xid/0rkgicew0xuv8ve-bwpwfp

Compute the original integral directly:

https://wolfram.com/xid/0rkgicew0xuv8ve-f66t23

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

https://wolfram.com/xid/0rkgicew0xuv8ve-9grqe

https://wolfram.com/xid/0rkgicew0xuv8ve-end738

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

https://wolfram.com/xid/0rkgicew0xuv8ve-db23hs

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

https://wolfram.com/xid/0rkgicew0xuv8ve-boflnz

https://wolfram.com/xid/0rkgicew0xuv8ve-ei12ro

Properties & Relations (3)Properties of the function, and connections to other functions
is bijective iff the equation
has exactly one solution for each
:

https://wolfram.com/xid/0rkgicew0xuv8ve-bgha30


https://wolfram.com/xid/0rkgicew0xuv8ve-fgftkt

Use Solve to find the solutions:

https://wolfram.com/xid/0rkgicew0xuv8ve-h9imdc


https://wolfram.com/xid/0rkgicew0xuv8ve-ec0f5e

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

https://wolfram.com/xid/0rkgicew0xuv8ve-b0d3so


https://wolfram.com/xid/0rkgicew0xuv8ve-mtrqj2


https://wolfram.com/xid/0rkgicew0xuv8ve-daqrl2

Use FunctionMonotonicity to determine the monotonicity of a function:

https://wolfram.com/xid/0rkgicew0xuv8ve-fmt43q


https://wolfram.com/xid/0rkgicew0xuv8ve-if28q4

Use Limit to compute the limits:

https://wolfram.com/xid/0rkgicew0xuv8ve-d2r6j0


https://wolfram.com/xid/0rkgicew0xuv8ve-bc9gqs

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

https://wolfram.com/xid/0rkgicew0xuv8ve-bfsvvm

https://wolfram.com/xid/0rkgicew0xuv8ve-cdw82t

Use Solve to find the polynomial inverse:

https://wolfram.com/xid/0rkgicew0xuv8ve-cqiub8

Verify that is a two-sided inverse of
:

https://wolfram.com/xid/0rkgicew0xuv8ve-cpx0gn


https://wolfram.com/xid/0rkgicew0xuv8ve-lbnzh

Possible Issues (2)Common pitfalls and unexpected behavior
FunctionBijective determines the real domain of functions using FunctionDomain:

https://wolfram.com/xid/0rkgicew0xuv8ve-c1rnh7

is bijective onto
in the real domain reported by FunctionDomain:

https://wolfram.com/xid/0rkgicew0xuv8ve-gfcxum


https://wolfram.com/xid/0rkgicew0xuv8ve-e806up

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

https://wolfram.com/xid/0rkgicew0xuv8ve-bhmdzf

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

https://wolfram.com/xid/0rkgicew0xuv8ve-h3a0uv

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

https://wolfram.com/xid/0rkgicew0xuv8ve-c7ycwd

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.
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
]}
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
]}