Injective, Surjective, and Bijective Functions

Understanding Function Mapping Properties

In mathematics, functions map elements from one set (the domain) to another set (the codomain). This mapping can be classified into several types based on how domain and codomain elements are connected. The three main types are injective, surjective, and bijective functions.

Let's assume we have a function $f: X \to Y$.

Injective Function (One-to-One)

A function is called injective or one-to-one if every distinct element in the domain $X$ maps to a distinct element in the codomain $Y$. In other words, no two different domain elements can have the same image (output) in the codomain.

Formal Definition:

A function $f: X \to Y$ is injective if for every $x_1, x_2 \in X$, the following holds:

Or, equivalently (using the contrapositive):

Analogy: Imagine every student in a class (domain) must have a unique student ID number (codomain). No two students can have the same ID number. The mapping function from students to ID numbers is an injective function.

Examples:

• The function $f(x) = 2x$ for $x \in \mathbb{R}$ is injective, because every distinct value of $x$ will produce a distinct $2x$.
• The function $g(x) = x^2$ for $x \in \mathbb{R}$ is not injective, because $g(2) = 4$ and $g(-2) = 4$. There are two different inputs ($2$ and $-2$) that produce the same output ($4$).

Surjective Function (Onto)

A function is called surjective or onto if every element in the codomain $Y$ is the image of at least one element in the domain $X$. In other words, there are no "unreachable" elements in the codomain that don't have a corresponding element in the domain. The range of a surjective function is equal to its codomain.

Formal Definition:

A function $f: X \to Y$ is surjective if for every $y \in Y$, there exists at least one $x \in X$ such that:

Analogy: Imagine every seat in a movie theater (codomain) must be occupied by at least one audience member (domain) when the movie starts. The mapping function from audience members to seats is surjective if all seats are filled.

Examples:

• The function $f(x) = x^3$ from $\mathbb{R} \to \mathbb{R}$ is surjective, because every real number $y$ in the codomain is the cube of some real number $x$ (specifically, $x = \sqrt[3]{y}$).
• The function $g(x) = x^2$ from $\mathbb{R} \to \mathbb{R}$ is not surjective, because there is no real number $x$ that yields $g(x) = -1$ (or any other negative number). Negative elements in the codomain have no corresponding element in the domain.
• However, if we restrict the codomain to $g(x) = x^2$ from $\mathbb{R} \to [0, \infty)$ (non-negative real numbers), then this function becomes surjective.

Bijective Function (One-to-One Correspondence)

A function is called bijective if it is both injective and surjective. This means that every element in the domain maps to a unique element in the codomain, and every element in the codomain has exactly one corresponding element in the domain.

A bijective function creates a perfect one-to-one correspondence between the elements of the domain and the codomain.

Formal Definition:

A function $f: X \to Y$ is bijective if for every $y \in Y$, there exists exactly one $x \in X$ such that:

Analogy: Imagine a perfect pairing between an equal number of men (domain) and women (codomain). Each man is paired with exactly one unique woman, and each woman is paired with exactly one unique man. This pairing function is bijective.

Important: A function can only have an inverse function if it is bijective.

Examples:

• The function $f(x) = 2x$ from $\mathbb{R} \to \mathbb{R}$ is bijective (injective and surjective).
• The function $f(x) = x^3$ from $\mathbb{R} \to \mathbb{R}$ is bijective (injective and surjective).
• The function $g(x) = x^2$ from $\mathbb{R} \to \mathbb{R}$ is not bijective (neither injective nor surjective).
• The function $h(x) = e^x$ from $\mathbb{R} \to \mathbb{R}$ is not bijective (injective but not surjective).
• The function $k(x) = x^3 - x$ from $\mathbb{R} \to \mathbb{R}$ is not bijective (surjective but not injective).