1.3 - Functions

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<?xml version="1.0" encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>1.3 - Functions</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>1.3 - Functions</h1> <h2>Definitions</h2> <dl> <dt><strong>Relation</strong></dt> <dd>A rule that associates a value in the domain with a value in the range.</dd> <dt><strong>Function</strong></dt> <dd>A function is a relation (rule) that assigns each element in the domain to exactly one element in the range.</dd> <dt><strong>Domain</strong></dt> <dd>The set of all the values which may be input into a function. That is, the set of all the values the independent variable may assume. Graphically, the domain is the set of all the x-coordinates.</dd> <dt><strong>Range</strong></dt> <dd>The set of all the values which are output when the function is evaluated at all the input values from the domain. That is, the set of all the values the dependent variable may assume. Graphically, the range is the set of all the y-coordinates.</dd> <dt><strong>Independent Variable</strong></dt> <dd>Typically, the independent variable is x. However, the independent variable is the variable which is free to assume different values independently of the other variable. Most of what we're going to do in this class will only involve one independent variable, but realize that it is possible to have more than one independent variable.</dd> <dt><strong>Dependent Variable</strong></dt> <dd>Typically, the dependent variable is y. However, the dependent variable is the variable which is determined based on the value of the independent variable(s). If a function is written as y=3x+2, then y depends on x, but x doesn't depend on y (in the form it's written in). So, x is independent of y, but y is dependent of x.</dd> <dt><strong>Implied Domain</strong></dt> <dd>The set of all real numbers for which the expression is defined.</dd> </dl> <h2>How is a function different from a relation?</h2> <p>Here are some guidelines for determining whether a relation is a function or not. </p> <ol> <li>Each and every element in the domain (x) must be matched with an element in the range (y).</li> <li>Every element in the domain (x) can only be matched with one element in the range (y).</li> <li>Different elements in the domain can go to the same element in the range. A y-coordinate may be repeated, but an x-coordinate may not.</li>
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<li>Some values in the range don't have to be used at all</li> </ol> <h2>Function Notation</h2> <p>Function notation is used to name functions for easy reference. Imagine if
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1.3 - Functions - &lt;?xml version=&quot;1.0&quot;...

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