In this section we’re going to review one of the more common functions in both calculus and the sciences.  However, before getting to this function let’s take a much more general approach to things.

Let’s start with , .  An exponential function is then a function in the form,

Note that we avoid  because that would give the constant function, .  We avoid  since this would also give a constant function and we avoid negative values of b for the following reason.  Let’s, for a second, suppose that we did allow b to be negative and look at the following function.

Let’s do some evaluation.

So, for some values of x we will get real numbers and for other values of x we well get complex numbers.  We want to avoid this and so if we require  this will not be a problem.

Let’s take a look at a couple of exponential functions.

This graph illustrates some very nice properties about exponential functions in general.

Properties of  

These will all be very useful properties to recall at times as we move throughout this course (and later Calculus courses for that matter…).

There is a very important exponential function that arises naturally in many places.  This function is called the natural exponential function.  However, for most people this is simply the exponential function.

So, since  we also know that  and .

Let’s take a quick look at an example.

The main point behind this problem is to make sure you can do this type of evaluation so make sure that you can get the values that we graphed in this example.  You will be asked to do this kind of evaluation on occasion in this class.

You will be seeing exponential functions in pretty much every chapter in this class so make sure that you are comfortable with them.