Numbers and Functions Steps into Analysis

Preliminary activity: make sure that you have access to graph-drawing facilities on a computer or graphic calculator, and that you can use these facilities with confidence.

Preliminary reading:Leavitt ch. 1.

Concurrent reading: Swann and Johnson, Hart, Reade, Smith, Spivak chs 4, 5, 6.

Further reading:Mason.

Before coming to university you will have worked with polynomials, trigonometric, logarithmic and exponential functions. Now we explore properties shared by all of these functions: continuity, differentiability and integrability.

Functions

When you read or hear the phrase ‘the function f(x)’, what comes to your mind? Perhaps a formula, perhaps a graph.

1 Write down what x can stand for, and what is meant by f, in the expression f(x). Compare your answer with the one in the summary at the end of this section.

We will introduce some special vocabulary in order to be clear what we mean when talking about functions.

The domain of a function

If f(x) = x2 and the values of x are 0, ±1, ±2, ±3, …, ±n, …, then the values of f(x) are 0, 1, 4, 9, …, n2, …. The set of possible values of x is called the domain of the function. When we say that x is a variable, we mean that the symbol x is being used to denote any member of the domain of a function. When the possible values of x are real numbers, the function is called a function of a real variable.

The range and co-domain of a function

The set of possible values of f(x) is called the range of the function, and any set which contains the range may be declared to be the co-domain of the function. We have just given a function with domain ℤ and range ℕ ∪ {0}, which we express symbolically by writing f: ℤ → ℕ ∪ {0}, with the definition f(x) = x2 (or f: x → x2).