## Exponential function

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.

 1. 2.

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