FunctionSurjective [f,x]
tests whether has at least one solution x in TemplateBox[{}, Reals] for each y∈Reals .
FunctionSurjective [f,x,dom]
tests whether has at least one solution x∈dom for each y∈dom.
FunctionSurjective [{f1,f2,…},{x1,x2,…},dom]
tests whether has at least one solution x1,x2,…∈dom for each y1,y2,…∈dom.
FunctionSurjective [{funs,xcons,ycons},xvars,yvars,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.
FunctionSurjective
FunctionSurjective [f,x]
tests whether has at least one solution x in TemplateBox[{}, Reals] for each y∈Reals .
FunctionSurjective [f,x,dom]
tests whether has at least one solution x∈dom for each y∈dom.
FunctionSurjective [{f1,f2,…},{x1,x2,…},dom]
tests whether has at least one solution x1,x2,…∈dom for each y1,y2,…∈dom.
FunctionSurjective [{funs,xcons,ycons},xvars,yvars,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:
-
- Possible settings for GenerateConditions include:
-
Automatic nongeneric conditions onlyTrue all conditionsFalse no conditionsNone return unevaluated if conditions are needed
- Possible settings for PerformanceGoal are "Speed" and "Quality".
Examples
open all close allBasic Examples (4)
Test surjectivity of a univariate function over the reals:
Test surjectivity over the complexes:
Test surjectivity of a polynomial mapping over the reals:
Test surjectivity of a polynomial with symbolic coefficients:
Scope (12)
Surjectivity over the reals:
Each value is attained at least once:
Surjectivity over a subset of the reals:
For positive , some values are not attained:
Surjectivity onto a subset of the reals:
Each positive value is for some positive :
Surjectivity over the complexes:
The value zero is not attained:
Surjectivity onto a subset of complexes:
Surjectivity over the integers:
Surjectivity of linear mappings:
A linear mapping is surjective iff the rank of its matrix is equal to the dimension of its codomain:
Surjectivity of polynomial mappings :
Each value is attained at least once:
This mapping is not surjective:
Some values are not attained:
Surjectivity of polynomial mappings :
Surjectivity of polynomial mappings :
Surjectivity of a real polynomial with symbolic parameters:
Surjectivity of a real polynomial mapping with symbolic parameters:
Options (4)
Assumptions (1)
FunctionSurjective gives a conditional answer here:
This checks the surjectivity for the remaining real values of :
GenerateConditions (2)
By default, FunctionSurjective may generate conditions on symbolic parameters:
With GenerateConditions None , FunctionSurjective fails instead of giving a conditional result:
This returns a conditionally valid result without stating the condition:
By default, all conditions are reported:
With GenerateConditions Automatic , conditions that are generically true are not reported:
PerformanceGoal (1)
Use PerformanceGoal to avoid potentially expensive computations:
The default setting uses all available techniques to try to produce a result:
Applications (11)
Basic Applications (7)
Check surjectivity of :
is surjective because it attains each value at least once:
Check surjectivity of :
is not surjective because the value is not attained:
is not surjective because it does not attain negative values:
The value is not attained:
is surjective as a function from TemplateBox[{}, NonNegativeReals] to TemplateBox[{}, NonNegativeReals]:
Each non-negative value is attained:
is surjective:
Each value is attained at least once:
TemplateBox[{x}, CosIntegral] is not surjective:
Some values, e.g. , are not attained:
A function is surjective if any horizontal line intersects its graph at least once:
If a horizontal line does not intersect the graph, the function is not surjective:
is bounded:
Bounded functions are not surjective:
If is continuous on and , then is surjective onto :
Use FunctionContinuous to check that is continuous in :
By the intermediate value theorem, restricted to is surjective onto :
An affine mapping is surjective if the rank of is equal to the number of rows of :
Solving Equations and Inequalities (1)
A function is surjective if the equation has at least one solution for any :
For each real , there is at least one real solution for :
Use Resolve to check the condition expressed using quantifiers:
Probability & Statistics (3)
A CDF for a continuous distribution is surjective onto the interval of probabilities (0,1) over its domain:
A SurvivalFunction for a continuous distribution is surjective onto the interval of probabilities (0,1) over its domain:
The quantile function Quantile for a distribution is surjective onto the domain of the distribution:
Properties & Relations (3)
is surjective iff the equation has at least one solution for each :
Use Solve to find the solutions:
A real continuous function on an interval is surjective iff the limits at endpoints are and :
Use Limit to compute the limits:
A function is surjective if its FunctionRange is True :
Possible Issues (1)
FunctionSurjective determines the real domain of functions using FunctionDomain :
is not surjective onto in the real domain reported by FunctionDomain :
is real valued over the whole reals and is surjective onto :
All subexpressions of need to be real valued for a point to belong to the real domain of :
Related Guides
History
Text
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.
APA
Wolfram Language. (2020). FunctionSurjective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSurjective.html
BibTeX
@misc{reference.wolfram_2025_functionsurjective, author="Wolfram Research", title="{FunctionSurjective}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionSurjective.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_functionsurjective, organization={Wolfram Research}, title={FunctionSurjective}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionSurjective.html}, note=[Accessed: 05-December-2025]}