A function whose rate of change is

proportional to its value exhibits

exponential growth if the constant of

proportionality is positive and

exponentional decay if the constant of

proportionality is negative. For

exponential growth, the function is

given by kbx with b > 1, and functions

governed by exponential decay are of

the same form with b < 1. Populations

might exhibit exponential growth in

the absence of constraints, while

quantities of a radioactive isotope

exhibit exponential decay.

If b > 1, the exponential function f(x)

= bx grows faster than any polynomial

(or rational) function. In other words,

if g(x) is a polynomial, there is some

positive number M such that

f(x) > g(x) for every x > M. Similarly, if

b < 1, the function bx has zero has a

horizontal asymptote for large

positive x and it nears this asymptote

faster than any rational function.