FunctionInjective is unable to decide injectivity of for arbitrary :

With assumed to be an odd integer, FunctionInjective succeeds:

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:

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

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:

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 :

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

SurvivalFunction and Quantile are injective as well:

Compute by change of variables:

If is an injective mapping, then :

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 :

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