Chapter1 - Section 1.1 Four Ways to Represent a Function...

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1 Section 1.1 Four Ways to Represent a Function • Goals – Learn to represent functions using • Words • Tables of values •G raphs •Fo rmu las – Present the Vertical Line Test for curves in the plane Section 1.1 Four Ways (cont’d) – Discuss piecewise defined functions , including the absolute value function – Study symmetry of functions, including even and odd functions – Discuss increasing and decreasing functions What is a Function? A function arises whenever one quantity depends on another. Here are four examples: A. The area A of a circle depends on the radius r of the circle. A and r are connected by the rule A = π r 2 . B. The population P of the world depends on the time t , even though the value of P ( t ) can only be estimated. In General… •A function is a rule that assigns… – to each element x in a set A – exactly one element, called f ( x ) , in a set B . • Usually we will take A and B to be sets of real numbers. Terminology • The set A is called the domain of the function. • The number f ( x ) is the value of f at x and is read “ f of x ”. • The range of f is the set of all possible values of f ( x ) as x varies throughout the domain. Terminology (cont’d) • A symbol that represents an arbitrary number in the domain of a function f is called an independent variable . • A symbol that represents a number in the range of f is called a dependent variable . – In the function A = π r 2 giving the area of a circle in terms of its radius, r is the independent variable, and A is the dependent variable.
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2 Machine Diagram • We can think of a function f as a machine: – If x is in the domain of f , then when x enters the machine, the machine produces an output according to the rule of the function. – So the domain is the set of all possible inputs and the range is the set of all possible outputs : Arrow Diagram • This is another way to picture a function. • Each arrow connects an element of A to an element of B . • The arrow indicates that… f ( x ) is associated with x , f ( a ) is associated with a , etc. • This is illustrated on the next slide: Arrow Diagram (cont’d) Graph of a Function • The graph of a function f is the set of ordered pairs • So the graph consists of all points ( x , y ) where y = f ( x ) and x is in the domain of f . • From the graph of f we can read… – the value of f ( x ) as being the height of the graph above the point x ; – the domain and range of f , as the next slide shows: () ( ) { } , |. xfx x A Graph (cont’d) Example The graph of a function f is shown at right. a) Find the values of f (1) and f (5) . b) What are the domain and range of f ?
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3 Example Sketch the graph and find the domain and range of each function: a) f ( x ) = 2 x –1 b)±± g ( x ) = x 2 •S o l u t i o n (See figures on the next slide.) a) The graph of f is a line with slope 2 and y -intercept –1 , so the domain and range of f are both R .
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This note was uploaded on 04/18/2008 for the course MATH 210 taught by Professor Zhoramanseur during the Spring '08 term at SUNY Oswego.

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Chapter1 - Section 1.1 Four Ways to Represent a Function...

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