Mapping an input to an output

Functions express the relation between an input value and an output value. Let's say we have two sets X and Y. A function f would map the relation of x \in X to exactly one y \in Y. Such a function posses defined properties, which can be used to classify them.

Image

So we could say y = f(x), which translates to f maps x to y, or y is the output of f when applied to x. Which intern is nothing else then saying: "The image of f : X \to Y is a set of all elements y \in Y such that y = f(x) for some x \in X."

This is written as: Im(f) = \{ f(x) | x \in X \} or: Im(f) = \{ y \in Y : \exists x \in X : y = f(x) \}

Target

Y is the target set of our function f. So this Im(f) \subset Y applies.

Domain

The definition set is the root set of elements. X is the definition set of our function f; which we write as Dom(f) = X.

Graph

The graph of a function is the set of all points (x, f(x)) in the coordinate plane.

Plotting a graph

Examples:

f: \mathbb{R} \to \mathbb{R},x \mapsto x^2 - 1 g: \mathbb{R}\backslash \{0\} \to \mathbb{R},x \mapsto \frac{1}{x}

![[Drawing 2025-05-26 17.27.54.excalidraw.svg]]

The image of f would be: \mathbb{R}_{\geq-1} And the image of g would be: \mathbb{R}_{\neq 0}

Operations

You can do math with function :).

It's pretty much what you would expect, but this \circ thingy is different/new. This denotes something like f \circ g = f(g(x)). First you apply g to x which would be y and then you apply f to y.