Examples for

Injectivity & Surjectivity

Injectivity and surjectivity describe properties of a function. An injection, or one-to-one function, is a function for which no two distinct inputs produce the same output. A surjection, or onto function, is a function for which every element in the codomain has at least one corresponding input in the domain which produces that output. A function that is both injective and surjective is called bijective. Wolfram|Alpha can determine whether a given function is injective and/or surjective over a specified domain.
Injectivity

Test if a function is an injection.

Determine whether a given function is injective:

is y=x^3+x a one-to-one function?
Is f(x) = x e^(-x^2) injective?

Determine injectivity on a specified domain:

is x^2 injective on x>0
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Surjectivity

Check if a function is a surjection.

Determine whether a given function is surjective:

is x^2-x surjective?
is g(x) = x^2 sin(x) an onto function

Determine surjectivity on a specified domain:

Is f(x)=(x^3 + x)/(x-2) for x<2 surjective?
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Bijectivity

Find out if a function is a bijection.

Determine whether a given function is bijective:

Is x^3 - x one-to-one and onto?

Determine bijectivity on a specified domain:

is tan(x) a bijection on (-pi/2,pi/2)
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