# Exponential functions (§6.3)

Recall that a **power function**
is a function *f*
of the form
for some constant *n*
called the **exponent** of the function.
In contrast, an **exponential function**
is a function *f*
of the form
for some constant *b* called the **base** of the function.
The base should be a positive number,
so that *b*^{x} makes sense for every real number *x*.
Recall that a **linear function**
is a function *f*
of the form

for some constants *m* and *b*.
Analogously, a **generalised exponential function**
is a function *f*
of the form
for some constants *b* and *C*.
If you don't remember any other values of a generalised exponential function,
remember these:

*f*(0) = *C**b*^{0} = *C*,
*f*(1) = *C**b*^{1} =
*C* · *b*,
*f*(−1) =
*C**b*^{−1} = *C*/*b*.

The domain of a generalised exponential function
is the set of all real numbers;
as long as *b* ≠ 1,
the range of the exponential function with that base
is the set of all *positive* numbers.
(Because *b* is positive, *b*^{x} is also positive.)
Then the range of a generalised exponential function
is the set of all real numbers with the same sign as *C*.
If *C* > 0 and *b* > 1,
then the generalised exponential function is increasing;
if either of these is reversed, then the function is decreasing;
if both are reversed, then it's increasing again.
(If *b* = 1 or *C* = 0,
then the exponential function is constant.)
Besides numbers such as 10, 2, and 1/2, which you are familiar with,
another common choice of base is a special number, about 2.72, known as e.
The importance of this number e will become clear when we look at applications.
Many calculators have a button
that calculates e^{x} from *x*;
in particular, you can calculate e itself, as e^{1}, using this button.

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