A function is called an *exponential function* if

When , is an increasing function
and
as
.
When , is a decreasing function
and
as
.
In either case, is a one-to-one function,
and has the horizontal asymptote .
The domain and the range of are given by
and
, respectively.

**Property of one-to-one functions.**
If is a one-to-one function,
implies .
In particular,
if
with positive real value ,
implies .

**Natural exponential function.**
A function is called the *natural exponential function* if

where
is the number given by (1).
**Exponential growth and decay.**
Suppose that the value at time is expressed as

with initial value at time .
If , we say that the value ``increases exponentially''
and is often called the ``rate of growth.''
If , we say that the value ``decreases exponentially''
and is often called the ``rate of decay.''

Department of Mathematics

*Last modified: 2005-09-29*