Let’s start off this section with the definition of an
exponential function.
Notice that the x
is now in the exponent and the base is a fixed number. This is exactly the opposite from what we’ve
seen to this point. To this point the
base has been the variable, x in most
cases, and the exponent was a fixed number.
However, despite these differences these functions evaluate in exactly
the same way as those that we are used to.
We will see some examples of exponential functions shortly.
Before we get too far into this section we should address
the restrictions on b. We avoid one and zero because in this case
the function would be,
and these are constant functions and won’t have many of the
same properties that general exponential functions have.
Next, we avoid negative numbers so that we don’t get any
complex values out of the function evaluation.
For instance if we allowed 
the function would be,
and as you can see there are some function evaluations that
will give complex numbers. We only want
real numbers to arise from function evaluation and so to make sure of this we
require that b not be a negative
number.
Now, let’s take a look at a couple of graphs. We will be able to get most of the properties
of exponential functions from these graphs.
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Example 1 Sketch
the graph of and on the same axis system.
Solution
Okay, since we don’t have any knowledge on what these
graphs look like we’re going to have to pick some values of x and do some function
evaluations. Function evaluation with
exponential functions works in exactly the same manner that all function
evaluation has worked to this point.
Whatever is in the parenthesis on the left we substitute into all the x’s on the right side.
Here are some evaluations for these two functions,
Here is the sketch of the two graphs.
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Note as well that we could have written in the following way,
Sometimes we’ll see this kind of exponential function and so
it’s important to be able to go between these two forms.
Now, let’s talk about some of the properties of exponential
functions.
Properties of
- The
graph of will always contain the point . Or put another way, regardless of the value of b.
- For
every possible b . Note that this implies that .
- If then the graph of will decrease as we move from left to
right. Check out the graph of above for verification of this
property.
- If then the graph of will increase as we move from left to
right. Check out the graph of above for verification of this
property.
- If then
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All of these properties except the final one can be verified
easily from the graphs in the first example.
We will hold off discussing the final property for a couple of sections
where we will actually be using it.
As a final topic in this section we need to discuss a
special exponential function. In fact
this is so special that for many people this is THE exponential function. Here it is,
where . Note the difference between and . In the first case b is any number that meets the restrictions given above while e is a very specific number. Also note that e is not a terminating decimal.
This special exponential function is very important and
arises naturally in many areas. As noted
above, this function arises so often that many people will think of this
function if you talk about exponential functions. We will see some of the applications of this
function in the final section of this chapter.
Let’s get a quick graph of this function.
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Example 2 Sketch
the graph of .
Solution
Let’s first build up a table of values for this function.
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x
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-2
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-1
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0
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1
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2
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f(x)
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0.1353…
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0.3679…
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1
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2.718…
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7.389…
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To get these evaluation (with the exception of ) you will need to use a
calculator. In fact, that is part of
the point of this example. Make sure that
you can run your calculator and verify these numbers.
Here is a sketch of this graph.
Notice that this is an increasing graph as we should
expect since .
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There is one final example that we need to work before
moving onto the next section. This
example is more about the evaluation process for exponential functions than the
graphing process. We need to be very
careful with the evaluation of exponential functions.
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Example 3 Sketch
the graph of .
Solution
Here is a quick table of values for this function.
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x
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-1
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0
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1
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2
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3
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g(x)
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32.945…
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9.591…
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1
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-2.161…
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-3.323…
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Now, as we stated above this example was more about the
evaluation process than the graph so let’s go through the first one to make
sure that you can do these.
Notice that when evaluating exponential functions we first
need to actually do the exponentiation before we multiply by any coefficients
(5 in this case). Also, we used only 3
decimal places here since we are only graphing. In many applications we will want to use
far more decimal places in these computations.
Here is a sketch of the graph.
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Notice that this graph violates all the properties we listed
above. That is okay. Those properties are only valid for functions
in the form or . We’ve got a lot more going on in this
function and so the properties, as written above, won’t hold for this function.