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10 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W...

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x y 4 3 2 1 1 2 -2 -1 ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 10 FUNCTIONS AND LINES 10.1. Functions and Graphs We shall be concerned with real valued functions of a real variable x . In other words, we study functions of the form f ( x ), where x R and f ( x ) is real valued. A convenient way to study and understand the properties of a function is to draw its graph. To do so, we make use of the xy -plane, and denote the values f ( x ) by using the y -axis. Then the graph of the function consists of all points ( x, f ( x )) for which the function is defined. Example 10.1.1. Consider the function f ( x ) = x 2 . This function is defined for all real values of x . For every x R , the value f ( x ) is real. We have f (0) = 0, f ( 1) = f (1) = 1 and f ( 2) = f (2) = 4. The graph of this function is given below. This chapter was written at Macquarie University in 1999.
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x y 4 3 2 1 1 2 x y x 0 x 1 y 1 y 0 P ( x 0 , y 0 ) Q ( x 1 , y 1 ) R ( x 1 , y 0 ) 10–2 W W L Chen and X T Duong : Elementary Mathematics Example 10.1.2. Consider the function f ( x ) = x . This function is defined for all non-negative real values of x but not defined for any negative values of x . For every non-negative x R , the value f ( x ) is real. We have f (0) = 0, f (1) = 1, f (2) = 2, f (3) = 3 and f (4) = 2. The graph of this function is given below. 10.2. Lines on the Plane In this section, we shall study the problem of lines and their graphs. Recall first of all that a line is determined if we know two of its points. Suppose that P ( x 0 , y 0 ) and Q ( x 1 , y 1 ) are two points on the xy -plane as shown in the picture below. We can consider the triangle PQR formed by P and Q as well as the point R ( x 1 , y 0 ). The length of the vertical side of this triangle is given by y 1 y 0 , while the length of the horizontal side of this triangle is given by x 1 x 0 . The ratio y 1 y 0 x 1 x 0 is called the slope of the line through the points P and Q .
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