## Exponential
Functions

An exponential
function has the form

The independent
variable x is the exponent and the constant b is the base of the exponential
function. While the constant _{a} can be any real number,
the base, *b*, must be positive and different from 1.

The domain
of an exponential function is the set of all real numbers. The range
is the set of all positive real numbers when a>0 and the set of negative
numbers when a<0. The graph looks like one of those shown below.

Notice that
the exponential function _{}
is increasing and concave up when a>0 and b>1, and decreasing
and concave up when a>0 and 0<b<1. If a<0, the graph is
reflected across the x- axis. The graph is decreasing and concave down
when a<0 and b>1 and increasing and concave down when a<0 and
0<b<1.

The graph
of an exponential function has no x-intercepts and exactly one y-intercept.

### Example

Graph the
function _{} and
find the y-intercept.

### Solution

The graph
is shown in Figure 3 below. To find the y-intercept, we replace x with
0 and evaluate f(0),

So, the y-intercept
is 1.8. Notice that the y- intercept is the same as the value of the
coefficient _{}.
This is always the case because _{}