Exponential function

A function $f$ is called an exponential function if

$\displaystyle f(x) = a^x .
$

When $ a > 1$, $f$ is an increasing function and $ f(x) \to 0$ as $ x \to -\infty$. When $ 0 < a < 1$, $f$ is a decreasing function and $ f(x) \to 0$ as $ x \to \infty$. In either case, $f$ is a one-to-one function, and has the horizontal asymptote $ y = 0$. The domain $ D$ and the range $ R$ of $f$ are given by $ D = (-\infty, \infty)$ and $ R = (0, \infty)$, respectively.

1. $ a > 1$   2. $ 0 < a < 1$
\includegraphics{lec07a.ps}   \includegraphics{lec07b.ps}

Property of one-to-one functions. If $f$ is a one-to-one function, $ f(x_1) = f(x_2)$ implies $ x_1 = x_2$. In particular, if $ f(x) = a^x$ with positive real value $ a \neq 1$, $ a^{x_1} = a^{x_2}$ implies $ x_1 = x_2$.

Natural exponential function. A function $f$ is called the natural exponential function if

$\displaystyle f(x) = e^x
$

where $ e \approx 2.71828\ldots$ is the number given by (1).

Exponential growth and decay. Suppose that the value $ q(t)$ at time $ t$ is expressed as

$\displaystyle q(t) = a e^{r t}
$

with initial value $a > 0$ at time $ t = 0$. If $ r > 0$, we say that the value $ q(t)$ ``increases exponentially'' and $r$ is often called the ``rate of growth.'' If $ r < 0$, we say that the value $ q(t)$ ``decreases exponentially'' and $r$ is often called the ``rate of decay.''



Department of Mathematics
Last modified: 2005-09-29