In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

Click 'View document' below to open Section 4 (5 pages, 104KB).

In Section 2 we describe how the graphs of polynomial and rational functions may be sketched by analysing their behaviour – for example, by using techniques of calculus. We assume that you are familiar with basic calculus and that its use is valid. In particular, we assume that the graphs of the functions under consideration consist of smooth curves.

Click 'View document' below to open Section 2 (16 pages, 200KB).

In Section 1 we formally define real functions and describe how they may arise when we try to solve equations. We remind you of some basic real functions and their graphs, and describe how some of the properties of these functions are featured in their graphs.

Click 'View document' below to open Section 1 (12 pages, 1.8MB).

A fundamental concept in mathematics is that of a function.

Consider, for example, the function f defined by

This is an example of a real function, because it associates with a given real number x the real number 2x2 − 1: it maps real numbers to real n
Author(s): The Open University

By the end of this section you should be able to:

Section 1: Real functions

• understand the definition of a real function;

• use the notation for intervals of the real line;

• recognise and use the graphs of the basic functions described in the audio section;

• understand the effect on a graph of translations, scalings, rotations and reflections;

• understand how the shape o
Author(s): The Open University

Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t
Author(s): The Open University

All written material contained within this unit originated at the Open University.

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

1. Join the 200,000 students currently studyi
Author(s): The Open University

Now try the quiz  and see if there are any areas you need to work on.

Author(s): The Open University

1 Use the method outlined in Example 9 to estimate each of the following, and then use yo
Author(s): The Open University

Scientific notation can be very useful when estimating the answers to calculations involving very large and/or small decimal numbers.

## Example 9

A lottery winner won £7851 000. He put the money straight into a deposit account which earns 7.5% interest per annum (i.e. each year). If he wanted to
Author(s): The Open University

1 Express each of the following numbers in scientific notation.

• (a) Light travels 9460 700 000 000 km in a year.

• (b) The average distance from the centre of the Earth to the centre o
Author(s): The Open University

Earlier you looked at place values for numbers, and why they were called powers of ten.