Exponential Functions

Rules of Exponents

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

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

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

Properties of the graph

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