Some notes for Quantitative Methods, lecture 1
Note: html isn't very good for typesetting maths. I use the following
notation in these notes: x^2 means "x squared", x^3 means "x cubed" and so on.
pi means 3.14159..., so that the area of a circle of radius r is written 2 pi r.
Lecture 1
- Overview of the course --- see elsewhere on this Web page.
- Functions: definition; domain; range; examples.
- Composition of functions.
- Even and odd functions.
Some definitions
- If a number y is completely determined by a number x, then y is a
function of x. This is written
y = f(x)
Example: Area of a circle = pi r2 where r is the radius.
- Set of all allowed x ("input") values is called the Domain of the
function.
- Set of all y ("output") values when x is in the domain, is called the
Range of the function.
Composition of functions
"Composition" is a fancy word for substituting one function into another, and
it is best illustrated by an example. Let f(x) = 3 x - 2 and g(x) = x + 1. Then
the composition of f and g, written (f o g)(x) or f(g(x)) is found by
replacing x in f, with g. Hence
(f o g)(x) = 3 g - 2 = 3 (x+1) - 2 = 3 x + 3 - 2 = 3 x + 1
We can also find (g o f)(x) using the same method:
(g o f)(x) = f + 1 = 3 x - 2 + 1 = 3 x - 1
Note that (f o g)(x) does not equal (g o f)(x).
Even and odd functions
Like plants and animals, functions can be classified into different
categories. We look here at even and odd functions; in the next lecture, we
shall mention periodic functions.
Functions can be classified as even, odd or neither. The definitions are
The function f(x) is even is f(x) = f(-x).
The function f(x) is odd is f(x) = -f(-x).
If f(x) has neither of these properties, then it is neither even nor odd.
As illustrated in the lecture, you can tell by looking at its graph whether a
function is even or odd. An even function looks exactly the same if you
reverse the x-axis --- turn it over. An odd function looks the same if you
rotate it about the origin by 180o. (This was easier to show with
an overhead projector than by trying to describe it in words.)
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Any constructive comments? E-mail me:
J.Deane@ee.surrey.ac.uk
J.Deane@ee.surrey.ac.uk
January 2001