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Unformatted text preview: Ch2/MATH1804/YMC/200910 1 Chapter 2. Functions 2.1. Functions and Graphs The idea of a function expresses the dependence between two quantities, one of which is given and the other is the output. A function associates a unique output with every input element. Definition 2.1 Let X , Y be sets. A function from X into Y is a rule which assigns EACH element in X to EXACTLY ONE element in Y . We call X and Y the domain and the codomain of the function respectively. If the function is denoted by f , then we may write f : X→ Y. When the domain and codmain of a function are both sets of real numbers, the function is said to be a realvalued function of one variable , and we write f : R→ R . From now on we shall only focus on realvalued functions of one variable (and realvalued functions of two variables in the final part of this course). Remark A variable that represents the input for a function is called an independent variable . A variable that represents the output for a function is called dependent variable because its value depends on the value of the independent variable. For example, if f ( x ) = x +1 for any x ∈ R , and if we set y = f ( x ) , then we have the equation y = x + 1 . In this case, the independent variable is x and the dependent variable is y . Graphs of Functions A way to visualize a function is its graph. If f is a realvalued function of one variable, its graph consists of the points in the Cartesian plane R 2 whose coordinates are the inputoutput pairs for f . In set notation, the graph is { ( x,y ) ∈ R 2 : x ∈ R , y = f ( x ) } . Example 2.2 In the figure below, (a) shows the graph of a typical function. The graph in (b) shows a function f : [2 , ∞ ) → R with domain [2 , ∞ ) . Note that the function is undefined for x < 2 . The graph in (c) does not represent a function since there are two different outputs for each input nonzero number x . Ch2/MATH1804/YMC/200910 2 Part (c) of the above example suggests that not every figure in R 2 represent the graph of a function. The key idea is that we cannot have two points with the same xcoordinate. This leads to the following vertical line test : The vertical line test If you have a figure and you would like to know whether it is the graph of a function, see whether any vertical line intersects the figure more than once. If so, it is NOT the graph of a function. But if there is no vertical line intersects the figure more than once, then the figure is indeed the graph of a function. Try to apply the vertical line test to the figures shown in Example 2.2. The following figure shows a circle of radius 1 centered at the origin. Is this figure the graph of a function? Ch2/MATH1804/YMC/200910 3 Here are some examples of simple functions: Special Functions (i) Constant Functions A function of the form f ( x ) = c , where c is a constant, is called a constant function . Let f ( x ) = 5 for any x ∈ R , then the output value is always equal to 5 . For example, f (2) = 5 ,...
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