# 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*

**(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 ...

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