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FunctionInjective

tests whether has at most one solution x∈Reals for each y.

tests whether has at most one solution x∈dom.

FunctionInjective[{f₁,f₂,…},{x₁,x₂,…},dom]

tests whether has at most one solution x₁,x₂,…∈dom.

FunctionInjective[{funs,xcons,ycons},xvars,yvars,dom]

tests whether has at most one solution with xvars∈dom restricted by the constraints xcons for each yvars∈dom restricted by the constraints ycons.

FunctionInjective

FunctionInjective[f,x]

tests whether has at most one solution x∈Reals for each y.

FunctionInjective[f,x,dom]

tests whether has at most one solution x∈dom.

FunctionInjective[{f₁,f₂,…},{x₁,x₂,…},dom]

tests whether has at most one solution x₁,x₂,…∈dom.

FunctionInjective[{funs,xcons,ycons},xvars,yvars,dom]

tests whether has at most one solution with xvars∈dom restricted by the constraints xcons for each yvars∈dom restricted by the constraints ycons.

Details and Options

An injective function is also known as one-to-one.
A function is injective if for each there is at most one such that .

FunctionInjective[{funs,xcons,ycons},xvars,yvars,dom] returns True if the mapping is injective, 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 injectivity of a univariate function over the reals:

Test injectivity over the complexes:

Test injectivity of a polynomial mapping over the reals:

Test injectivity of a polynomial with symbolic coefficients:

Scope (12)

Injectivity over the reals:

Some values are attained more than once:

Injectivity over a subset of the reals:

For , each value is attained at most once:

Injectivity over the inverse image of a subset of the reals:

Each value with is attained at most once:

Injectivity over the complexes:

Each value is attained at most once:

Injectivity over a subset of complexes:

is not injective over the whole complex plane:

Some values are attained more than once:

Injectivity over the integers:

Injectivity of linear mappings:

A linear mapping is injective iff the rank of its matrix is equal to the dimension of its domain:

Injectivity of polynomial mappings :

Each value is attained at most once:

This mapping is not injective:

Some values are attained more than once:

Injectivity of polynomial mappings :

The mapping, equal to the real and imaginary parts of , is injective in the non-negative quadrant:

Injectivity of polynomial mappings :

The Jacobian determinant of an injective 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 injective:

Injectivity of a real polynomial with symbolic parameters:

Injectivity of a real polynomial mapping with symbolic parameters:

Options (4)

Assumptions (1)

FunctionInjective is unable to decide injectivity of for arbitrary :

With assumed to be an odd integer, FunctionInjective succeeds:

GenerateConditions (2)

By default, FunctionInjective may generate conditions on symbolic parameters:

With GenerateConditionsNone, FunctionInjective 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 (17)

Basic Applications (8)

Check injectivity of :

is not injective, because the value is attained more than once:

Check injectivity of :

Each value is attained at most once:

Check injectivity of in its real domain:

is injective in its real domain :

Check injectivity of Clip[x] over unrestricted reals:

Values and are attained more than once on, respectively, [-∞,-1] and [1, ∞]:

Clip[x] restricted to the interval [-1, 1] is injective:

A function is injective if any horizontal line intersects its graph at most once:

If a horizontal line intersects the graph more than once, the function is not injective:

Periodic functions are not injective:

Use FunctionPeriod to check whether a function is periodic:

A periodic function attains each value infinitely many times:

Functions whose derivative has a fixed positive or negative sign are injective:

Use FunctionSign to find the sign of a derivative:

The function is strictly increasing, and hence injective:

Integrals of functions that have a fixed positive or negative sign are injective:

Integrate a positive piecewise function for argument values in [0,2]:

The integral is injective for 0≤x≤2:

Injective functions do not need to be continuous or monotonic:

A continuous injective function does not need to be monotonic:

The function is continuous, but its domain is not a connected set:

Check injectivity of a mapping :

A part of the ParametricPlot has been covered twice:

For positive and , the mapping is injective:

Each point of the ParametricPlot is covered once:

Here, the point {0,0} is clearly covered multiple times:

These are all the and that map to {0,0}:

Apart from that, the mapping is injective:

Solving Equations & Inequalities (4)

is injective if for any value of , the equation has at most one solution for :

There is one solution of :

There are no solutions of :

There is at most one solution of :

The mapping is not injective:

The equation has two solutions:

The restriction of to is injective:

The equation has at most one solution with :

Any injective function has an inverse function:

The domain of the inverse function is the range of :

A differentiable function with a nonzero derivative, defined on a connected set, is injective:

The derivative of is positive for :

is injective for :

The inverse function of satisfies the equation :

Check that the result is indeed the inverse function of for :

Probability & Statistics (2)

The CDF of a distribution with a strictly positive PDF is injective:

SurvivalFunction and Quantile are injective as well:

Calculus (3)

Compute by change of variables:

If is an injective mapping, then $int_Uf(g(u)) TemplateBox[{TemplateBox[{{{(, {partial, g}, )}, /, {(, {partial, u}, )}}}, Det]}, Abs]du=int_(g(U))f(v)dv$ :