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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
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Assumptions  
GenerateConditions  
PerformanceGoal  
Applications  
Basic Applications  
Solving Equations and Inequalities  
Probability & Statistics  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page

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:
  • 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

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Basic 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 GenerateConditionsNone, 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 GenerateConditionsAutomatic, 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 :

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.

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: 30-October-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: 30-October-2025]}