Surjective Functions (and a Proof!) | Surjections, Onto Functions, Surjective Proofs

What is a surjection? A surjection, also called a surjective function or onto function, is a special type of function with an interesting property. We’ll define surjective functions, go over a method of how to prove a function is surjective, and see some interesting examples in today’s video math lesson on surjections!

How to prove a function is surjective?    • How to Prove a Function is Surjective | Lo...  

A function f that maps A to B is surjective if and only if, for all y in B, there exists x in A such that f(x) = y. In other words, a function is surjective if every element in the codomain gets mapped to by at least one element in the domain!

Don’t confuse surjections with injections! They’re very different. An injective function is a function where every element in the codomain gets mapped to AT MOST one time, so no two domain elements map to the same codomain element. Check out my lesson on injections:    • Injective Functions (and a Proof!) | Injec...  

SOLUTION TO PRACTICE PROBLEM:

Let f map R (the reals) to R and be defined by f(x) = (1/2)*(x^3 + 5). Take y in R. Then, (2y - 5)^(1/3) is also a real number because 2y - 5 is real by closure and the cube root is defined for all real numbers. Notice,

f( (2y - 5)^(1/3) ) = (1/2)*( [(2y - 5)^(1/3)]^3 + 5 ) = (1/2)*(2y - 5 + 5) = y.

Thus, f is surjective.

Note the expression (2y - 5)^(1/3) came from solving f(x) = y for x, to find the domain element that would map to y.



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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