4.1 - Exponential Functions and Their Graphs

4.1 - Exponential Functions and Their Graphs -

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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>4.1 - Exponential Functions and Their Graphs </title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>4.1 - Exponential Functions and Their Graphs</h1> <h2>Exponential Functions</h2> <p>So far, we have been dealing with algebraic functions. Algebraic functions are functions which can be expressed using arithmetic operations and whose values are either rational or a root of a rational number. Now, we will be dealing with transcendental functions. Transcendental functions return values which may not be expressible as rational numbers or roots of rational numbers. </p> <p>Algebraic equations can be solved most of the time by hand. Transcendental functions can often be solved by hand with a calculator necessary if you want a decimal approximation. However when transcendental and algebraic functions are mixed in an equation, graphical or numerical techniques are sometimes the only way to find the solution. </p> <p>The simplest exponential function is: f(x) = a<sup>x</sup>, a>0, a≠1 </p> <p>The reasons for the restrictions are simple. If a&le;0, then when you raise it to a rational power, you may not get a real number. Example: If a=-2, then (-2)<sup>0.5</sup> = sqrt(- 2) which isn't real. If a=1, then no matter what x is, the value of f(x) is 1. That is a pretty boring function, and it is certainly not one-to-one. </p> <p>Recall that one-to-one functions had several properties that make them desirable. They have inverses that are also functions. They can be applied to both sides of an equation.</p> <h2>Graphs of Exponential Functions</h2> <p><img src="2x.gif" alt="y=2^x" width="218" height="277" class="imgrt" />The graph of y=2<sup>x</sup> is shown to the right. Here are some properties of the exponential function when the base is greater than 1. </p>
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This note was uploaded on 10/18/2011 for the course MAT 1033 taught by Professor Brown during the Spring '10 term at Valencia.

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4.1 - Exponential Functions and Their Graphs -

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