FunctionInjective

FunctionInjective[f,x]

tests whether has at most one solution xReals for each y.

FunctionInjective[f,x,dom]

tests whether has at most one solution xdom.

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 xvarsdom restricted by the constraints xcons for each yvarsdom 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$Assumptionsassumptions on parameters
    GenerateConditionsTruewhether to generate conditions on parameters
    PerformanceGoal$PerformanceGoalwhether to prioritize speed or quality
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic 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 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 TemplateBox[{n, x}, BesselI] for arbitrary :

With assumed to be an odd integer, FunctionInjective succeeds:

GenerateConditions  (2)

By default, FunctionInjective may generate conditions on symbolic parameters:

With GenerateConditionsNone, 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 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  (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 0x2:

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:

Find the range of to determine the integration bounds:

Compute the integral:

Compute the original integral directly:

Compute the surface area of a ball with radius using a rational parametrization:

Check that the parametrization is injective:

The surface area is equal to the integral of square root of Gram determinant of :

Integrate over the "eight surface" using a rational parametrization:

Check that the parametrization is injective outside a one-dimensional set:

Compute the integral of g(f(t,u))sqrt(TemplateBox[{{TemplateBox[{{(, {{(, {partial, f}, )}, /, {(, {partial, {{, {t, ,, u}, }}}, )}}, )}}, Transpose], ., {(, {{(, {partial, f}, )}, /, {(, {partial, {{, {t, ,, u}, }}}, )}}, )}}}, Det]):

Compare with the value computed using the implicit description of the "eight surface":

Properties & Relations  (3)

is injective iff the equation has at most one solution for each :

Use Solve to find the solutions:

A real univariate continuous function on a connected set is injective iff it is monotonic:

Use FunctionMonotonicity to determine the monotonicity of a function:

A complex polynomial mapping is injective iff it has a polynomial inverse:

Use Solve to find the polynomial inverse:

Verify that is a two-sided inverse of :

Possible Issues  (1)

FunctionInjective determines the real domain of functions using FunctionDomain:

is injective in the real domain reported by FunctionDomain:

is real valued and not injective over the whole reals:

All subexpressions of need to be real valued for a point to belong to the real domain of :

Wolfram Research (2020), FunctionInjective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionInjective.html.

Text

Wolfram Research (2020), FunctionInjective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionInjective.html.

BibTeX

@misc{reference.wolfram_2021_functioninjective, author="Wolfram Research", title="{FunctionInjective}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionInjective.html}", note=[Accessed: 26-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_functioninjective, organization={Wolfram Research}, title={FunctionInjective}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionInjective.html}, note=[Accessed: 26-October-2021 ]}

CMS

Wolfram Language. 2020. "FunctionInjective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionInjective.html.

APA

Wolfram Language. (2020). FunctionInjective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionInjective.html