Exponential Functions

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Rules of Exponents

 If s, t, a, b are real numbers with a, b ${\displaystyle >}$ 0, then
${\displaystyle a^{s}\cdot a^{t}=a^{s+t}~(a^{s})^{t}=a^{st}~(ab)^{s}=a^{s}b^{s}1^{s}=1~a^{-s}={\frac {1}{a^{s}}}=\left({\frac {1}{a}}\right)^{s}~a^{0}=1}$


Now that we can define an exponential function: ${\displaystyle f(x)=Ca^{x}}$ 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 ${\displaystyle f(x)=Ca^{x}}$ is an exponential function,

then ${\displaystyle {\frac {f(x+1)}{f(x)}}=a}$

Properties of the graph

 Properties of ${\displaystyle f(x)=a^{x},~a>1}$
1. The domain is ${\displaystyle (-\infty ,\infty )}$ and the range is ${\displaystyle (0,\infty )}$
2. The y-intercept is (0, 1) and there is no x-intercept.
3. The x-axis is a horizontal asymptote
4. ${\displaystyle f(x)}$ is an increasing, one-to-one function
5. The graph contains the three points ${\displaystyle (0,1),~(1,a),~(-1,{\frac {1}{a}})}$
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 ${\displaystyle f(x)=a^{x},~0
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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