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	$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

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Basic Examples (4)

Test bijectivity of a univariate function over the reals:

Test bijectivity over the complexes:

Test bijectivity of a polynomial mapping over the reals:

Test bijectivity of a polynomial with symbolic coefficients:

Scope (10)

Bijectivity over the reals:

Some values are attained more than once:

Bijectivity between subsets of the reals:

For , each positive value is attained exactly once:

Bijectivity over the complexes:

Some values are not attained:

The logarithm is bijective onto :

Bijectivity over a subset of complexes:

is not bijective over the whole complex plane:

Some values are attained more than once:

Bijectivity over the integers:

Bijectivity of linear mappings:

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

Bijectivity of polynomial mappings :

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

Bijectivity of polynomial mappings :

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

The Jacobian conjecture states that the reverse implication is true:

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

Bijectivity of a real polynomial with symbolic parameters:

Bijectivity of a real polynomial mapping with symbolic parameters:

Options (4)

Assumptions (1)

FunctionBijective gives a conditional answer here:

This checks the bijectivity for the remaining real values of :

GenerateConditions (2)

By default, FunctionBijective may generate conditions on symbolic parameters:

With GenerateConditions->None, FunctionBijective 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 (8)

Check bijectivity of :

Each value is attained exactly once:

is not bijective:

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

is not bijective:

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

is bijective in its real domain:

Each value is attained exactly once:

Show that is bijective:

Show that is not bijective:

A function is bijective iff it is injective and surjective:

is not bijective because it is not injective:

is not bijective because it is not surjective:

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

is not bijective:

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

Composition of bijective functions is bijective:

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

Probability (1)

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

SurvivalFunction is bijective onto as well:

Quantile is bijective onto the reals:

Calculus (2)

Compute by change of variables:

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