FunctionSurjective
✖
FunctionSurjective
tests whether has at least one solution x1,x2,…∈dom for each y1,y2,…∈dom.
tests whether has at least one solution with xvars∈dom restricted by the constraints xcons for each yvars∈dom restricted by the constraints ycons.
Details and Options
- A surjective function is also known as onto or an onto mapping.
- A function is surjective if for every there is at least one such that .
- 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.
- FunctionSurjective[{funs,xcons,ycons},xvars,yvars,dom] returns True if the mapping is surjective, where is the solution set of xcons and is the solution set of ycons.
- 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 surjectivity of a univariate function over the reals:
https://wolfram.com/xid/08umqdsvoau4aui-b2fcqx
Test surjectivity over the complexes:
https://wolfram.com/xid/08umqdsvoau4aui-0dl46
Test surjectivity of a polynomial mapping over the reals:
https://wolfram.com/xid/08umqdsvoau4aui-dp3h13
Test surjectivity of a polynomial with symbolic coefficients:
https://wolfram.com/xid/08umqdsvoau4aui-kqrk3c
Scope (12)Survey of the scope of standard use cases
https://wolfram.com/xid/08umqdsvoau4aui-gxhl11
Each value is attained at least once:
https://wolfram.com/xid/08umqdsvoau4aui-d5ri0g
Surjectivity over a subset of the reals:
https://wolfram.com/xid/08umqdsvoau4aui-glvby
For positive , some values are not attained:
https://wolfram.com/xid/08umqdsvoau4aui-h18c7
Surjectivity onto a subset of the reals:
https://wolfram.com/xid/08umqdsvoau4aui-d66d8y
Each positive value is for some positive :
https://wolfram.com/xid/08umqdsvoau4aui-ctca0g
Surjectivity over the complexes:
https://wolfram.com/xid/08umqdsvoau4aui-gb8vpl
The value zero is not attained:
https://wolfram.com/xid/08umqdsvoau4aui-unv0t
Surjectivity onto a subset of complexes:
https://wolfram.com/xid/08umqdsvoau4aui-e5opda
Surjectivity over the integers:
https://wolfram.com/xid/08umqdsvoau4aui-oypp7s
Surjectivity of linear mappings:
https://wolfram.com/xid/08umqdsvoau4aui-fuu3lu
https://wolfram.com/xid/08umqdsvoau4aui-dw1lhm
https://wolfram.com/xid/08umqdsvoau4aui-bb3g0a
https://wolfram.com/xid/08umqdsvoau4aui-huze44
A linear mapping is surjective iff the rank of its matrix is equal to the dimension of its codomain:
https://wolfram.com/xid/08umqdsvoau4aui-dwvhdm
Surjectivity of polynomial mappings :
https://wolfram.com/xid/08umqdsvoau4aui-en7wnr
Each value is attained at least once:
https://wolfram.com/xid/08umqdsvoau4aui-bccrwp
This mapping is not surjective:
https://wolfram.com/xid/08umqdsvoau4aui-hdmdbu
https://wolfram.com/xid/08umqdsvoau4aui-hh23j9
Surjectivity of polynomial mappings :
https://wolfram.com/xid/08umqdsvoau4aui-dcji69
Surjectivity of polynomial mappings :
https://wolfram.com/xid/08umqdsvoau4aui-dcz40b
Surjectivity of a real polynomial with symbolic parameters:
https://wolfram.com/xid/08umqdsvoau4aui-dqtdtb
Surjectivity of a real polynomial mapping with symbolic parameters:
https://wolfram.com/xid/08umqdsvoau4aui-yju7
Options (4)Common values & functionality for each option
Assumptions (1)
FunctionSurjective gives a conditional answer here:
https://wolfram.com/xid/08umqdsvoau4aui-bnv0jw
This checks the surjectivity for the remaining real values of :
https://wolfram.com/xid/08umqdsvoau4aui-c50qtk
GenerateConditions (2)
By default, FunctionSurjective may generate conditions on symbolic parameters:
https://wolfram.com/xid/08umqdsvoau4aui-osy2z
With GenerateConditionsNone, FunctionSurjective fails instead of giving a conditional result:
https://wolfram.com/xid/08umqdsvoau4aui-fe9ubz
This returns a conditionally valid result without stating the condition:
https://wolfram.com/xid/08umqdsvoau4aui-na5ydu
By default, all conditions are reported:
https://wolfram.com/xid/08umqdsvoau4aui-tdcquw
With GenerateConditionsAutomatic, conditions that are generically true are not reported:
https://wolfram.com/xid/08umqdsvoau4aui-291b1m
PerformanceGoal (1)
Use PerformanceGoal to avoid potentially expensive computations:
https://wolfram.com/xid/08umqdsvoau4aui-i86kxj
The default setting uses all available techniques to try to produce a result:
https://wolfram.com/xid/08umqdsvoau4aui-i9gq
Applications (11)Sample problems that can be solved with this function
Basic Applications (7)
https://wolfram.com/xid/08umqdsvoau4aui-tmkzs
is surjective because it attains each value at least once:
https://wolfram.com/xid/08umqdsvoau4aui-hwemwk
https://wolfram.com/xid/08umqdsvoau4aui-5p94k
is not surjective because the value is not attained:
https://wolfram.com/xid/08umqdsvoau4aui-dob2rz
is not surjective because it does not attain negative values:
https://wolfram.com/xid/08umqdsvoau4aui-rz87j
https://wolfram.com/xid/08umqdsvoau4aui-fhmver
is surjective as a function from to :
https://wolfram.com/xid/08umqdsvoau4aui-kz0n6
Each non-negative value is attained:
https://wolfram.com/xid/08umqdsvoau4aui-g0ztcj
https://wolfram.com/xid/08umqdsvoau4aui-g639dg
Each value is attained at least once:
https://wolfram.com/xid/08umqdsvoau4aui-smrt9
https://wolfram.com/xid/08umqdsvoau4aui-b1wh39
Some values, e.g. , are not attained:
https://wolfram.com/xid/08umqdsvoau4aui-eidcke
A function is surjective if any horizontal line intersects its graph at least once:
https://wolfram.com/xid/08umqdsvoau4aui-bethas
https://wolfram.com/xid/08umqdsvoau4aui-djdklc
If a horizontal line does not intersect the graph, the function is not surjective:
https://wolfram.com/xid/08umqdsvoau4aui-hybo03
https://wolfram.com/xid/08umqdsvoau4aui-p3tdpb
https://wolfram.com/xid/08umqdsvoau4aui-mi5ep
Bounded functions are not surjective:
https://wolfram.com/xid/08umqdsvoau4aui-hfe8tb
If is continuous on and , then is surjective onto :
https://wolfram.com/xid/08umqdsvoau4aui-k79eri
https://wolfram.com/xid/08umqdsvoau4aui-i5xwge
Use FunctionContinuous to check that is continuous in :
https://wolfram.com/xid/08umqdsvoau4aui-gwhxce
By the intermediate value theorem, restricted to is surjective onto :
https://wolfram.com/xid/08umqdsvoau4aui-chr9a
https://wolfram.com/xid/08umqdsvoau4aui-bx6mvx
An affine mapping is surjective if the rank of is equal to the number of rows of :
https://wolfram.com/xid/08umqdsvoau4aui-m9q295
https://wolfram.com/xid/08umqdsvoau4aui-hq1fwk
https://wolfram.com/xid/08umqdsvoau4aui-bet2ie
https://wolfram.com/xid/08umqdsvoau4aui-fclhyh
https://wolfram.com/xid/08umqdsvoau4aui-d64ko
https://wolfram.com/xid/08umqdsvoau4aui-cp7s2y
Solving Equations and Inequalities (1)
A function is surjective if the equation has at least one solution for any :
https://wolfram.com/xid/08umqdsvoau4aui-c2h13o
https://wolfram.com/xid/08umqdsvoau4aui-fsca99
For each real , there is at least one real solution for :
https://wolfram.com/xid/08umqdsvoau4aui-fnkkdu
Use Resolve to check the condition expressed using quantifiers:
https://wolfram.com/xid/08umqdsvoau4aui-dvrfk3
https://wolfram.com/xid/08umqdsvoau4aui-k28m21
Probability & Statistics (3)
A CDF for a continuous distribution is surjective onto the interval of probabilities (0,1) over its domain:
https://wolfram.com/xid/08umqdsvoau4aui-l16efo
https://wolfram.com/xid/08umqdsvoau4aui-bybrxp
https://wolfram.com/xid/08umqdsvoau4aui-e654vt
A SurvivalFunction for a continuous distribution is surjective onto the interval of probabilities (0,1) over its domain:
https://wolfram.com/xid/08umqdsvoau4aui-fvov3m
https://wolfram.com/xid/08umqdsvoau4aui-oxfmxd
https://wolfram.com/xid/08umqdsvoau4aui-bgw7z
The quantile function Quantile for a distribution is surjective onto the domain of the distribution:
https://wolfram.com/xid/08umqdsvoau4aui-bas5y6
https://wolfram.com/xid/08umqdsvoau4aui-bp1vf1
https://wolfram.com/xid/08umqdsvoau4aui-c3w06j
Properties & Relations (3)Properties of the function, and connections to other functions
is surjective iff the equation has at least one solution for each :
https://wolfram.com/xid/08umqdsvoau4aui-bgha30
https://wolfram.com/xid/08umqdsvoau4aui-fgftkt
Use Solve to find the solutions:
https://wolfram.com/xid/08umqdsvoau4aui-h9imdc
https://wolfram.com/xid/08umqdsvoau4aui-ec0f5e
A real continuous function on an interval is surjective iff the limits at endpoints are and :
https://wolfram.com/xid/08umqdsvoau4aui-b0d3so
https://wolfram.com/xid/08umqdsvoau4aui-mtrqj2
https://wolfram.com/xid/08umqdsvoau4aui-daqrl2
Use Limit to compute the limits:
https://wolfram.com/xid/08umqdsvoau4aui-fmt43q
https://wolfram.com/xid/08umqdsvoau4aui-if28q4
A function is surjective if its FunctionRange is True:
https://wolfram.com/xid/08umqdsvoau4aui-jwh4ee
https://wolfram.com/xid/08umqdsvoau4aui-e842wa
Possible Issues (1)Common pitfalls and unexpected behavior
FunctionSurjective determines the real domain of functions using FunctionDomain:
https://wolfram.com/xid/08umqdsvoau4aui-c1rnh7
is not surjective onto in the real domain reported by FunctionDomain:
https://wolfram.com/xid/08umqdsvoau4aui-gfcxum
https://wolfram.com/xid/08umqdsvoau4aui-e806up
is real valued over the whole reals and is surjective onto :
https://wolfram.com/xid/08umqdsvoau4aui-bhmdzf
All subexpressions of need to be real valued for a point to belong to the real domain of :
https://wolfram.com/xid/08umqdsvoau4aui-h3a0uv
Wolfram Research (2020), FunctionSurjective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSurjective.html.
Text
Wolfram Research (2020), FunctionSurjective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSurjective.html.
Wolfram Research (2020), FunctionSurjective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSurjective.html.
CMS
Wolfram Language. 2020. "FunctionSurjective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionSurjective.html.
Wolfram Language. 2020. "FunctionSurjective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionSurjective.html.
APA
Wolfram Language. (2020). FunctionSurjective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSurjective.html
Wolfram Language. (2020). FunctionSurjective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSurjective.html
BibTeX
@misc{reference.wolfram_2024_functionsurjective, author="Wolfram Research", title="{FunctionSurjective}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionSurjective.html}", note=[Accessed: 09-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_functionsurjective, organization={Wolfram Research}, title={FunctionSurjective}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionSurjective.html}, note=[Accessed: 09-January-2025
]}