Functions

Sources:

  1. Bernard Kolman, Robert C. Busby & Sharon Cutler Ros. (2014). Functions. Discrete Mathematical Structures (6th ed., pp. 206-246). Pearson.
  2. John Hubbard; Barbara Burke Hubbard. (2015). Vector calculus, linear algebra, and differential forms: A unified approach (5th ed., pp. 9-11). Matrix Editions.

In this section we define the notion of a function, a special type of relation.

Functions

Definition (Function). Let \(A\) and \(B\) be nonempty sets. A function \(f\) from \(A\) to \(B\), which is denoted \(f: A \rightarrow B\), is a relation from \(A\) to \(B\) such that for every \(a \in A\), there exists a unique \(b \in B\) such that \(f(a) = b\).

  • Recalling from the definition of relation, \(A\) and \(B\) are \(\operatorname{Dom}(f)\), \(\operatorname{Ran}(f)\), seperately.

  • Naturally, if \(a\) is not in \(\operatorname{Dom}(f)\), then \(f(a)=\varnothing\). If \(f(a)=\{b\}\), it is traditional to identify the set \(\{b\}\) with the element \(b\) and write \(f(a)=b\). We will follow this custom, since no confusion results. The relation \(f\) can then be described as the set of pairs \(\{(a, f(a)) \mid a \in \operatorname{Dom}(f)\}\).

this is not a function

The above relation is NOT a function: Not well defined at a, not defined at b.

Images

Definition (Image). The set of all values of \(f\) is called its image: \(b\) is an element of the image of a function \(f: A \rightarrow B\) if there exists an \(a \in A\) such that \(f(a)=b\).

  • The codomain of a function may be considerably larger than its image. For example, the image of the squaring function \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x)=x^2\) is the nonnegative real numbers; the codomain is \(\mathbb{R}\).