Graphing Exponential Functions

We now define exponential functions. We assume that ax has meaning for any real number x and any positive real number a and that the laws of exponents still hold, though we will not prove them here.

We require the base to be positive in order to avoid the imaginary numbers that would occur by taking even roots of negative numbers—an example is (1)1/2, the square root of −1, which is not a real number. The restriction a1 is made to exclude the constant function f(x)=1x=1, which does not have an inverse that is a function because it is not one-to-one.

The following are examples of exponential ...

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