Exponential Functions

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Rules of Exponents

 If s, t, a, b are real numbers with a, b  0, then
 

Now that we can define an exponential function: where a is a positive number, that is not 1, and C is a nonzero number. Then f(x) is an exponential function. We call c the initial value, because if x is a variable for time, f(0) = C.


Properties

The first thing we note, is if is an exponential function,

then


Properties of the graph

 Properties of 
 1. The domain is  and the range is 
 2. The y-intercept is (0, 1) and there is no x-intercept.
 3. The x-axis is a horizontal asymptote
 4.  is an increasing, one-to-one function
 5. The graph contains the three points 
 6. The graph of f is smooth and continuous. (Here smooth means you can take as many derivatives as you want)

Note: You do not have to worry about what it means for a function to be smooth, or what a derivative is, until calculus.


 Properties of 
 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.


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