# FunctionBijective

FunctionBijective[f,x]

tests whether has exactly one solution xReals for each yReals.

FunctionBijective[f,x,dom]

tests whether has exactly one solution xdom for each ydom.

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 xvarsdom restricted by the constraints xcons for each yvarsdom 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

open allclose all

## 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 , 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 , 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 , 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 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 :

Check that is a bijective mapping :

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 bijective onto the ball less a lower-dimensional set:

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

## Properties & Relations(3) is bijective iff the equation has exactly one solution for each :

Use Solve to find the solutions:

A real continuous function on an interval is bijective iff it is monotonic and the limits at endpoints are and :

Use FunctionMonotonicity to determine the monotonicity of a function:

Use Limit to compute the limits:

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

Use Solve to find the polynomial inverse:

Verify that is a two-sided inverse of :

## Possible Issues(2)

FunctionBijective determines the real domain of functions using FunctionDomain: is bijective onto in the real domain reported by FunctionDomain: is real valued and not bijective onto over the whole reals:

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

FunctionBijective restricts the domain to the inverse image of the solution set of ycons:

Division by two is a bijection between the even integers (the inverse image of integers) and the integers.