Algebra and Trigonometry, 5th Edition

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.

Exponential Function

The function $f(x)=a^x$, where x is a real number, $a>0$ and $a\ne 1$, is called the exponential function, base a.

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 ...