chapter1 - Chapter 1. Functions: Limits and Continuity 1.1...

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Chapter 1. Functions: Limits and Continuity 1.1 Functions It is common that the values of one variable depend on the values of another. E.g. the area A of a re- gion on the plane enclosed by a circle depends on the radius r of the circle ( A = πr 2 , r > 0 . ) Many years ago, the Swiss mathematician Euler invented the symbol y = f ( x ) to denote the statement that y is a function of x ”. A function represents a rule that assigns a unique value y to each value x. We refer to x as the independent variable and y the dependent variable.
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2 MA1505 Chapter 1. Functions: Limits and Continuity One can also think of a function as an input-output system/process: input the value x and output the value y = f ( x ) . (This becomes particularly useful when we combine or composite functions together.) Unless otherwise stated, in this course, we are only concerned with real values (or real numbers). The whole collection of real numbers is denoted by R . So for our functions, the values of x and y belong to R . 1.1.1 Domain and Range There may be constraints on the possible values of x , e.g. y = 1 /x , where we require that x 6 = 0 . The collection D in which x takes values is called the domain of the function f . 2
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3 MA1505 Chapter 1. Functions: Limits and Continuity Symbolically, we write f : D -→ R x 7-→ y = f ( x ) . The R appearing on the right side of the arrow in the above notation is called the codomain of f . It indicates that the output values of this function are real numbers. On the other hand, the actual collection of y values, where y = f ( x ) and x is allowed to take the values in D , is known as the range of f (denoted by R .) E.g. the range of the function f = 1 /x is given by R = R - { 0 } . 3
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4 MA1505 Chapter 1. Functions: Limits and Continuity 1.1.2 Example The function A : R 0 R ; r 7→ πr 2 (also written as A ( r ) = πr 2 ) gives the area of a circle as a function of its radius; e.g., A (2) = 4 π . It is clear that the range of this function is the set of all nonnegative numbers R 0 . What about A ( - 2) =?. This is not defined since - 2 lies outside the domain of A . 1.1.3 Interval Notation For the next few examples, we shall use the interval notation . Let a and b be two real numbers with a < b . Then the interval notation refers to the following: 4
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5 MA1505 Chapter 1. Functions: Limits and Continuity [ a,b ] = { x R | a x b } (this set is called the closed interval from a to b ) ( a,b ) = { x R | a < x < b } (this set is called the open interval from a to b ) [ a,b ) = {
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This note was uploaded on 11/25/2010 for the course MATHS MA1505 taught by Professor Liu during the Spring '09 term at National University of Singapore.

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chapter1 - Chapter 1. Functions: Limits and Continuity 1.1...

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