An Exponential Function with base b is a function of the form:
f(x) = bx, where b > 0, b6= 1 is a real number.
We know the meaning of brif ris a rational number. What if ris irrational? What we
do is we approximate the value of brby using rational approximate for r. For example,
to approximate 5π, we may approximate it as 53.12, 53.141 , 53.1415, 53.14159.... In advance
mathematics one can define the value of 5πto be the limit of these approximations. For
now, just realize that such approximation can be used to approximate the value of brfor
any positive number band any real number r.
Since bxcan be defined for all real numbers r, the domain of an exponential function is all
real numbers.
The range of bxis all real numbers greater than 0.
Note the difference between an exponential function and a power function.
x2is a power function. In this function, the exponent is the constant.
2xis an exponential function. In this function, the base is the constant but the exponent
is the variable (input).
An exponential function is always positive. And if in addition 0 <b<1, fis a decreasing
function. That is f(x) decreases as xincreases.
If b > 1, then fis an increasing function. I.e. f(x) increases in value as xincreases. In
fact, an increasing exponential function (with any base) increases faster than any polynomial
function.
While the property of an exponential function with an base b > 1 is the same, one particular
useful base is the number e∼2.718281828 · · ·. The exponential function definied by f(x) =
exis the natural exponential function.
