Basic Technical Mathematics, 11th Edition

13.1 Exponential Functions

• The Exponential Function • Graphing Exponential Functions • Features of Exponential Functions

In Chapter 11, we showed that any rational number can be used as an exponent. Now letting the exponent be a variable, we define the exponential function as

$$y\hspace{0.17em}=\hspace{0.17em}{b}^x$$ (13.1)

where $b\hspace{0.17em}>\hspace{0.17em}0\hspace{0.17em},\hspace{0.17em}b\hspace{0.17em}\ne \hspace{0.17em}1\hspace{0.17em},\hspace{0.17em}$ and x is any real number. The number b is called the base.

For an exponential function, we use only real numbers. Therefore, $b\hspace{0.17em}>\hspace{0.17em}0\hspace{0.17em},\hspace{0.17em}$ because if b were negative and x were a fractional exponent with an even-number denominator, y would be imaginary. Also, $b\hspace{0.17em}\ne \hspace{0.17em}1\hspace{0.17em},\hspace{0.17em}$ since 1 to any real power is 1 (y would be constant).

EXAMPLE 1 Exponential functions

From the definition, $y\hspace{0.17em}=\hspace{0.17em}{3}^x$ is an exponential function, but $y\hspace{0.17em}=\hspace{0.17em}{(\hspace{0.17em}-\hspace{0.17em}3)}^x$ is not because ...