# Graphing Exponential Functions

We now define exponential functions. We assume that ${a}^{x}$ 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 $a\ne 1$ is made to exclude the constant function $f(x)={1}^{x}=1$, 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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