Chapter 4 – Functions

1. INTRODUCTION

As with “set” the concept of “function” is of fundamental importance in mathematics. It permeates all branches of the subject from elementary algebra to quantum mechanics. The following discussion will be restricted to functions defined on sets of real numbers only.

1.1 Definitions and Examples

We are all familiar with the idea of the dependence of the value of one quantity, y, on the value of another, x, usually determined by a formula.

For example, the value of the area y of a square depends of the value x of the common length of each of its sides, and is determined by the formula y = x2.

e.g. if x = 2, y = 22 = 4 and if x = 6, y = 62 = 36

Mathematicians view the above dependence of y values on x values as the “function” y = x2 and mean a mapping which links a number x to its square, x2, which we might write as x  x2.

In this example x does not have to be a nice positive integer like 2 or 6 since we can “work out” x2 for every real number x:

(-5/2)2 = 25/4 so -5/2  25/4

(3)2 = 3 so 3  3

We will say the function y = x2 is defined for all real numbers x, or equivalently, the domain of the function y = x2 is the set R of all real numbers.

This is not always the case. We cannot evaluate (i.e. work out) 1/x when x = 0, although we can for any other real number x. Consequently, the domain of the function y = 1/x is the set of all real numbers except x = 0.

Using the notation of set difference (chapter 1, section 2) this can be writtern R – {0}, but is more usually written R \ {0}.

Similarly, y = (x – 3) is defined for all x  3 only and so the domain of this function is
{ x : x  R, x  3 }.
Notice that in all the above examples, whenever a value is chosen from the domain of the function, the formula generates a unique value for y.

This is an important feature of functions.

We can now make some formal definitions.

Definition 4.1

A function f from a non-empty set X to a non-empty set Y is a rule which associates with every element x  X a unique element y  Y.

X is called the domain of the function and Y is called the codomain.

Notation

We usually use small letters f, g, … to denote functions.

A function from X to Y is written f : X  Y.

The (unique) element y  Y associated with the element x  X is written f(x) and is called the value of f at x or the image of x under f.

We often write x  f(x) (i.e. x  y = f(x) ) or simply y = f(x) when defining a function.

Definition 4.2

The set of all elements of Y which are images of elements of X is called the range of f, and written range (f).

So range (f) = { y : y  Y, y = f(x), x  X } and clearly range (f)  codomain (f).

Example 4.1 Let X = { a, b, c } and Y = { p, q, r, s }.

We can define a function f : X  Y by selecting the elements of X one by one and associating with each a unique element of Y.

For example, f(a) = q , f(b) = s and f(c) = p is a valid function definition,

and range (f) = { q, s, p }  Y.

Note that the same function could be defined by a diagram:

For such a diagram to be valid function definition there must be one, and only one, arrow from every element x  X.

Example 4.2 Let A = { a, b, c } , B = { d, e }

The diagram

properly defines a function f : A  B since every element of A has a unique arrow from it to an element of B.

Here f(a) = d , f(b) = e and f(c) = e , and

range (f) = { d, e } = B.

Example 4.3 Neither of the diagrams

(i) (ii)

provide proper function definitions since
(i) f(b) is not defined (ii) f(c) is not uniquely defined.

Note: The diagrams of examples 4.1, 4.2 and 4.3 are called set mapping diagrams.
Example 4.4 We can define a function f : R  R by f(x) = 1 + 4x.

This is a proper definition of a function since 1 + 4x is uniquely determined for every real number x.

It is easy to see that any number y  R is an image