Up to this point we have limited our study to constant exponents. Now we will explore functions with variable exponents.

**Evaluate**.

**Step 1**: Choose any values for

*x*. Choose some negative values, zero, and some positive values.

**Step 2**: Evaluate to find the corresponding

*y*-values.

**Step 3**: Plot the points. The points you plot the better the graph will look.

Notice that the

*y*-values can never actually attain zero. They get infinitely close when*x*is negative so the graph will be asymptotic to the*x*-axis. In other words, the line*y*= 0 is a horizontal asymptote.**Instructional Video**: Graph Exponential Functions

*x*,

*g*(

*x*) increases more rapidly than

*f*(

*x*). Also the domain and range for the two functions are the same. In addition, they are both asymptotic to the

*x*-axis (or

*y*= 0) and they both have the same

*y*-intercept (0, 1). Now we will explore exponential functions with bases greater than zero but less than 1.

**Tip**: All exponential functions of the form

*y*=

*b*^

*x*have (0, 1) as a

*y*-intercept, no

*x*-intercept, and the

*x*-axis will be a horizontal asymptote.

Looking at the two exponential two previous exponential functions on the same set of axes we will be able compare their rates of growth.

*g*(

*x*) decreases more rapidly than

*f*(

*x*). Also the domain, range, and

*y*-intercept for the two functions are the same.

**Tip:**All exponential functions of the form

*y*=

*b*^(

*x*+

*h*) +

*k*have a horizontal asymptote at

*y*=

*k*.

It is also useful to note the rigid transformations in the above graphs. In the previous example the basic graph of

*y*= 3^*x*was shifted down two units and to the left 1 unit.Notice that all of the exponential functions graphed above pass the horizontal line test. Therefore we may conclude that they are all one-to-one and have an inverse. This is one of the most important observations of this section. We will define the inverse of these functions in subsequent sections.

A common error when stating the range is to include the

*y*-value that defines the asymptote. In the previous problem one might think the range is [ 2, inf ) but this is incorrect. The

*y*-values get infinitely close to 2 but never actually attain that value – this is what it means to be asymptotic. So the range is ( 2, inf ).

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