Functions and Graphs

A function or mapping is a relationship between one set and another. For each element of the first set (which is called the domain) an element of the second set (called the range) is assigned. We write

f : X → Y,

to mean that the function named f maps elements in the domain X to elements in the range Y, where X and Y are sets. Signals are modeled as functions. So far, we have concentrated only on defining the domain and the range of a signal, and have not talked much about the function itself as a mathematical object. We will now show that a function is itself a set. We will then talk about sets of functions (sets of signals), and functions on those sets (systems).

If the domain of f is finite, f can be defined using a table. For example, the function Score that maps names into scores on an exam can be given by

*name*	Score(name)
JohnBrown	90.0
JaneLiu	90.4
…	…

Each row of the table has a member x ∈ X in the domain and a member y ∈ Y in the range. Thus, each row of the table is an element

(x, y) ∈ X × Y.

The set of all these elements (one for each row) fully defines the function. It is called the graph of the function, and is given formally by

graph( f ) = {(x, f(x)) | x ∈ X }

Notice that

graph( f ) ⊂ X × Y

Also notice that this definition works even if the domain is not finite, although the tabular representation does not.

It is not accidental that this set is called the graph of the function. Consider the audio signal shown below:

If you had a java-enabled browser, you would see an applet here.

This plot represents a function Voice: [0, 1] → Int16. , where Int16 is the set of 16-bit integers. In fact, it is a very literal depiction of graph(Voice). The rectangle outlining the plot is the set [0, 1] × Int16. A red dot is placed at each (x, y) in this set that is a member of the subset graph(Voice).

Week 2

Functions and Graphs