3.4 Injective, surjective, and bijective functions

Functions are like matchmakers, pairing elements from one set with another. Injective functions ensure no two elements share a match. Surjective functions make sure everyone in the second set gets paired up.

Bijective functions are the perfect matchmakers. They create unique pairs between two sets, leaving no one out. These special functions can be "undone" with inverse functions, reversing the pairing process.

Injective Functions

Definition and Properties

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• Injective function maps distinct elements in the domain to distinct elements in the codomain
• Also known as a one-to-one function because each element in the codomain is paired with at most one element in the domain
• Formally, a function $f: A \to B$ is injective if for all $a_1, a_2 \in A$, if $f(a_1) = f(b_1)$, then $a_1 = a_2$
• Injective functions preserve distinctness and do not collapse or combine elements

Determining Injectivity

• To determine if a function is injective, check if any two distinct elements in the domain map to the same element in the codomain
• Graphically, a function is injective if every horizontal line intersects the graph at most once (horizontal line test)
• For functions defined by formulas, injective functions typically have a unique $x$ variable on the right-hand side of the equation
• Examples of injective functions include [f(x) = 2x + 1](https://www.fiveableKeyTerm:f(x)_=_2x_+_1), [g(x) = e^x](https://www.fiveableKeyTerm:g(x)_=_e^x), and the identity function [h(x) = x](https://www.fiveableKeyTerm:h(x)_=_x)

Surjective Functions

Definition and Properties

• Surjective function maps the domain onto the entire codomain, meaning every element in the codomain is mapped to by at least one element in the domain
• Also known as an onto function because the range of the function is equal to the codomain
• Formally, a function $f: A \to B$ is surjective if for every $b \in B$, there exists an $a \in A$ such that $f(a) = b$
• Surjective functions ensure that no element in the codomain is "left out" or unmapped

Determining Surjectivity

• To determine if a function is surjective, check if every element in the codomain is mapped to by at least one element in the domain
• Graphically, a function is surjective if every horizontal line intersects the graph at least once
• For functions defined by formulas, surjective functions typically have the codomain expressed in terms of the function and the domain
• Examples of surjective functions include [f(x) = x^2](https://www.fiveableKeyTerm:f(x)_=_x^2) from $\mathbb{R}$ to $[0, \infty)$, and $g(x) = \arctan(x)$ from $\mathbb{R}$ to $(-\frac{\pi}{2}, \frac{\pi}{2})$

Bijective Functions and Invertibility

Definition and Properties of Bijective Functions

• Bijective function is both injective and surjective, meaning it is a one-to-one correspondence between the domain and codomain
• Each element in the domain is paired with exactly one element in the codomain, and each element in the codomain is paired with exactly one element in the domain
• Formally, a function $f: A \to B$ is bijective if it is both injective and surjective
• Bijective functions are perfect "pairing functions" that preserve distinctness and cover the entire codomain

Invertibility and Inverse Functions

• Bijective functions are invertible, meaning they have a unique inverse function that "undoes" the original function
• The inverse function, denoted as $f^{-1}$, maps each element in the codomain back to its unique corresponding element in the domain
• For a bijective function $f: A \to B$, the inverse function $f^{-1}: B \to A$ satisfies $f^{-1}(f(a)) = a$ for all $a \in A$ and $f(f^{-1}(b)) = b$ for all $b \in B$
• To find the inverse of a bijective function defined by a formula, switch $x$ and $y$, and then solve for $y$
• Examples of bijective functions and their inverses include $f(x) = 2x + 1$ with $f^{-1}(x) = \frac{x-1}{2}$, and $g(x) = e^x$ with $g^{-1}(x) = \ln(x)$