https://wiki.math.ucr.edu/index.php?title=Exponential_Functions&feed=atom&action=history Exponential Functions - Revision history 2022-05-21T03:13:23Z Revision history for this page on the wiki MediaWiki 1.35.0 https://wiki.math.ucr.edu/index.php?title=Exponential_Functions&diff=1131&oldid=prev MathAdmin: Created page with "<div class="noautonum">__TOC__</div> ==Rules of Exponents== If s, t, a, b are real numbers with a, b $>$ 0, then [itex]a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{..." 2015-10-23T04:28:56Z <p>Created page with &quot;&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt; ==Rules of Exponents== If s, t, a, b are real numbers with a, b &lt;math&gt; &gt; &lt;/math&gt; 0, then &lt;math&gt;a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{...&quot;</p> <p><b>New page</b></p><div>&lt;div class=&quot;noautonum&quot;&gt;__TOC__&lt;/div&gt;<br /> ==Rules of Exponents==<br /> <br /> If s, t, a, b are real numbers with a, b &lt;math&gt; &gt; &lt;/math&gt; 0, then<br /> &lt;math&gt;a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{st}~(ab)^s = a^sb^s<br /> 1^s = 1~a^{-s} = \frac{1}{a^s} = \left(\frac{1}{a}\right)^s~a^0 = 1&lt;/math&gt;<br /> <br /> Now that we can define an exponential function:<br /> &lt;math&gt;f(x) = Ca^x&lt;/math&gt; where a is a positive number, that is not 1, and C is a nonzero number.<br /> Then f(x) is an exponential function. We call c the initial value, because if x is a variable for time,<br /> f(0) = C. <br /> <br /> <br /> ==Properties==<br /> <br /> The first thing we note, is if &lt;math&gt;f(x) = Ca^x&lt;/math&gt; is an exponential function,<br /> <br /> then &lt;math&gt;\frac{f(x+1)}{f(x)} = a&lt;/math&gt;<br /> <br /> <br /> ==Properties of the graph==<br /> <br /> Properties of &lt;math&gt;f(x) = a^x,~ a &gt; 1&lt;/math&gt;<br /> 1. The domain is &lt;math&gt;(-\infty, \infty)&lt;/math&gt; and the range is &lt;math&gt;(0, \infty)&lt;/math&gt;<br /> 2. The y-intercept is (0, 1) and there is no x-intercept.<br /> 3. The x-axis is a horizontal asymptote<br /> 4. &lt;math&gt; f(x)&lt;/math&gt; is an increasing, one-to-one function<br /> 5. The graph contains the three points &lt;math&gt;(0, 1),~(1, a),~(-1, \frac{1}{a})&lt;/math&gt;<br /> 6. The graph of f is smooth and continuous. (Here smooth means you can take as many derivatives as you want)<br /> <br /> Note: You do not have to worry about what it means for a function to be smooth, or what a derivative is, until calculus.<br /> <br /> <br /> Properties of &lt;math&gt;f(x) = a^x, ~ 0 &lt; a &lt; 1 &lt;/math&gt;<br /> 1. This type of exponential function has the same properties as the one above EXCEPT in property 4, f(x) is decreasing instead of increasing.<br /> <br /> <br /> <br /> [[Math_5|'''Return to Topics Page]]</div> MathAdmin