Functions

Description

Let \(A\) and \(B\) be nonempty sets. A function \(f\) from \(A\) to \(B\) is an assignment of exactly one element of \(B\) to each element of \(A\). We write \(f(a) = b\) if \(b\) is the unique element of \(B\) assigned by the function \(f\) to the element \(a\) of \(A\). If \(f\) is a function from \(A\) to \(B\), we write \(f: A \rightarrow B\)

Result of Set Under a Function

Image

Let \(f\) be a function from \(A\) to \(B\) and let \(S\) be a subset of \(A\). The image of \(S\) under the function \(f\) is the subset of \(B\) that consists of the images of the elements of \(S\). We denote the image of \(S\) by \(f(S)\), so

\(f(S) = \{ t \mid \exists s \in S, \ t = f(s) \}\)

We also use the shorthand \(\{ f(s) \mid s \in S \}\) to denote this set.

The image of a function is the set of all output values it may produce.

\(f\) is a function from domain \(X\) to codomain \(Y\). The yellow oval inside \(Y\) is the image of \(f\).

Preimage

For a given function, the set of all elements of the domain that are mapped into a given subset of the codomain.

\(f^{-1}(Y)\)

Function Types

One to One Function (Injective)

A function \(f\) is said to be one-to-one, or an injection, if and only if \(f(x) = f(y)\) implies that \(x = y\) for all \(x\) and \(y\) in the domain of \(f\).

A function is said to be injective if it is one-to-one.

Increasing and Decreasing Function

A function \(f\) whose domain and codomain are subsets of the set of real numbers is called increasing if \(f(x) ≤ f(y)\), and strictly increasing if \(f(x) < f(y)\), whenever \(x < y\) and \(x\) and \(y\) are in the domain of \(f\). Similarly, \(f\) is called decreasing if \(f(x) ≥ f(y)\), and strictly decreasing if \(f(x) > f(y)\), whenever \(x < y\) and \(x\) and \(y\) are in the domain of \(f\). (The word strictly in this definition indicates a strict inequality.)

Onto Function (Surjective)

A function \(f\) from \(X\) to \(Y\) is called onto, or a surjection, if and only if for every element \(y \in Y\) there is at least an element \(x \in X\) with \(f(x) = y\).

A function \(f\) is called surjective if it is onto.

Identity Function

Is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality \(f(X) = X\) is true for all values of \(X\) to which \(f\) can be applied.

We will show this function by \(I\) (capital \(i\)).

Invertible Function

A function is invertible if and only if is a onto and one-to-one function.

Composition

Let \(g\) be a function from the set \(A\) to the set \(B\) and let \(f\) be a function from the set \(B\) to the set \(C\). The composition of the functions \(f\) and \(g\), denoted for all \(a \in A\) by \(f \circ g\), is the function from \(A\) to \(C\) defined by

\((f \circ g)(a) = f(g(a))\)