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.

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

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