# 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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