Surjective Functions

Surjectivity ensures that a function "hits" every possible target value. In functional equations, proving surjectivity often allows us to substitute variables with specific target values (like 0).

Definition

Definition (Surjective / Onto)

A function

f : A \to B

is surjective if for every element

y \in B

, there exists at least one element

x \in A

such that

f (x) = y

Range (f) = Codomain (f)

Proving Surjectivity

A common tactic is to show that the range of

f

is "large enough" (e.g., covers all real numbers).

Example

Prove that if

f (x + f (y)) = f (x) + y

, then

f

is surjective.

Proof. Fix $x = 0$ . The equation becomes $f (f (y)) = f (0) + y$ . The right side, $f (0) + y$ , is a linear function of $y$ that covers all real numbers as $y$ varies. For any target $z \in R$ , we can choose $y = z - f (0)$ . Then $f (f (y)) = z$ . Since $z$ is an output of $f$ (specifically, at input $f (y)$ ), $f$ is surjective. ∎

Properties

If $f : R \to R$ is a continuous function such that $lim_{x \to \infty} f (x) = \infty$ and $lim_{x \to - \infty} f (x) = - \infty$ , then $f$ is surjective (by Intermediate Value Theorem).
Polynomials of odd degree are surjective over $R$ .

Practice Problems

Exercise (Problem 1)

Is the function

f (x) = e^{x}

surjective from

R \to R

? What about

R \to (0, \infty)

Exercise (Problem 2)

Prove that

f (x) = x^{3} - x

is surjective over

R

Exercise (Problem 3)

Given

f : Z \to Z

defined by

f (n) = 2 n

, determine if it is surjective.