FunctionBijective
FunctionBijective[f,x]
tests whether has exactly one solution x∈Reals for each y∈Reals.
FunctionBijective[f,x,dom]
tests whether has exactly one solution x∈dom for each y∈dom.
FunctionBijective[{f_{1},f_{2},…},{x_{1},x_{2},…},dom]
tests whether has exactly one solution x_{1},x_{2},…∈dom for each y_{1},y_{2},…∈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 onetoone 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 allBasic Examples (4)
Scope (10)
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:
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 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)
Each value is attained exactly once:
Some values, e.g. , are attained more than once:
Some values, e.g. , are not attained:
is bijective in its real domain:
Each value is attained exactly once:
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:
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 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 lowerdimensional 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:
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.
Text
Wolfram Research (2020), FunctionBijective, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionBijective.html.
CMS
Wolfram Language. 2020. "FunctionBijective." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionBijective.html.
APA
Wolfram Language. (2020). FunctionBijective. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionBijective.html