Function Properties: One-To-One Correspondence

Definition. [One-To-One Correspondence Functions] A function is called bijective (also known as one-to-one correspondence) if it is both injective and surjective.

This means every element in the codomain is mapped to by exactly one element from the domain, and vice versa.

For example, the function \( f(x) = x + 5 \) from \( \mathbb{R} \to \mathbb{R} \) is bijective, since it is both injective and surjective.

From the previous slides, we also now know that $f(x) = 2x + 3$ is a bijection, while $f(x) = x^2$ isn't.

In bijective functions, there is a perfect pairing between domain and codomain elements.

Only bijective functions have well-defined inverses: we'll cover inverses on the next slide.