A **power function**
is a function *f*
of the form

*f*(*x*) =*x*^{n},

*f*(*x*) =*b*^{x},

Now recall that a **linear function**
is a function *f*
of the form

*f*(*x*) =*m**x*+*b*,

*f*(*x*) =*C**b*^{x},

If you don't remember any other values of a generalized 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 generalized 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 generalized exponential function
is the set of all real numbers with the same sign as *C*.
(But if *b* = 1 or *C* = 0,
then the range consists of only *C*.)
If *C* > 0 and *b* > 1,
then the generalized exponential function is increasing;
if either of these is reversed, then the function is decreasing;
if both are reversed, then it's increasing again.
(But if *b* = 1 or *C* = 0,
then the 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.

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2021 September 24. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`https://tobybartels.name/MATH-1300/2024FA/expfn/`

.