2009
11.02

## Exponential functions

The function $\emph{f}$ defined by $f(x) = b^{x}$ where $b$, and the exponent $x$ is any real number, is called an exponential function with base $b$

When you work with exponential functions, it may be necessary to apply rules for exponents. These rules are as follows, where $m$ and $n$ are real numbers and $a$ and $b$ are positive.

• $a^{m} \cdot a^{n} = a^{m+n}$
• $\frac{a^{m}}{a^{n}} = a^{m-n}$
• $(a^{m})^{n} = a^{m \cdot n}$
• $(ab)^{n} = a^{n} b^{n}$
• $(\frac{a}{b})^{n} = \frac{a^{n}}{b^{n}}$
• $(a)^{1} = a$
• $a^{0} = 1$
• $a^{-n} = \frac{1}{a^{n}}$

Graphs of some exponential functions are shown below:

plot .1^x, .5^x, .99^x, (1 + x)^(1/x), 2^x, 4^x, 8^x

Here, some properties of exponential functions

• The domain of an exponential function is all real numbers
• The range is all positive real numbers
• Since $b^{0} = 1$ for every base $b$, each graph has y-intercept (0, 1). There is no x-intercept.
• If $b > 0$, then $y = b^{x}$ is an increasing function; its graph rises from left to right. Also

$\mathop {\lim }\limits_{ x \to \infty } b^{x} = \infty$ and $\mathop {\lim }\limits_{ x \to -\infty } b^{x} = 0$

• Thus as $x \rightarrow -\infty$, the graph has the x-axis as a horizontal asymptote.
• If $0 < b < 1$, then $y = b^{x}$ is a decreasing function; its graph falls from left to right. Also

$\mathop {\lim }\limits_{ x \to \infty } b^{x} = 0$ and $\mathop {\lim }\limits_{ x \to -\infty } b^{x} = \infty$

• Thus as $x \rightarrow \infty$, the graph has the x-axis as the horizontal asymptote.