Lane ORCCA (2020–2021): Open Resources for Community College Algebra

Subsection 14.1.1 Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.

Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.

Our definition says \(b\ne 1\text{.}\) If we let \(b=1\text{,}\) then \(f\left(x\right)={b}^{x}\) becomes \(f\left(x\right)={1}^{x}\text{.}\) Since \({1}^{x}=1\) for all real numbers, \(f\left(x\right)={1}^{}\text{.}\) This is the constant function.

Our definition also says \(b>0\text{.}\) If we let a base be negative, say \(-4\text{,}\) then \(f\left(x\right)={\left(-4\right)}^{x}\) is not a real number when \(x=\frac{1}{2}\text{:}\)

In fact, \(f\left(x\right)={\left(-4\right)}^{x}\) would not be a real number any time \(x\) is a fraction with an even denominator. So our definition requires \(b>0\text{.}\)

By graphing a few exponential functions, we will be able to see their unique properties.

Example 14.1.3.

On the same coordinate system graph \(f\left(x\right)={2}^{x}\) and \(g\left(x\right)={3}^{x}\text{.}\)

Explanation

We will use point plotting to graph the functions.

Checkpoint 14.1.6.

Graph: \(f\left(x\right)={4}^{x}\text{.}\)

Explanation

Checkpoint 14.1.8.

Graph: \(g\left(x\right)={5}^{x}\text{.}\)

Explanation

If we look at the graphs from the previous examples, we can identify some of the properties of exponential functions.

The graphs of \(f\left(x\right)={2}^{x}\) and \(g\left(x\right)={3}^{x}\text{,}\) as well as the graphs of \(f\left(x\right)={4}^{x}\) and \(g\left(x\right)={5}^{x}\text{,}\) all have the same basic shape. This is the shape we expect from an exponential function where \(b>1\text{.}\)

We notice, that for each function, the graph contains the point \(\left(0,1\right)\text{.}\) This makes sense because \({b}^{0}=1\) for any \(b>1\text{.}\)

The graph of each function, \(f\left(x\right)={b}^{x}\) also contains the point \(\left(1,b\right)\text{.}\) The graph of \(f\left(x\right)={2}^{x}\) contains \(\left(1,2\right)\) and the graph of \(g\left(x\right)={3}^{x}\) contains \(\left(1,3\right)\text{.}\) This makes sense as \({b}^{1}=b\) for any \(b>1\text{.}\)

Notice too, that the graph of each function \(f\left(x\right)={b}^{x}\) also contains the point \(\left(-1,\frac{1}{b}\right)\text{.}\) The graph of \(f\left(x\right)={2}^{x}\) contains \(\left(-1,\frac{1}{2}\right)\) and the graph of \(g\left(x\right)={3}^{x}\) contains \(\left(-1,\frac{1}{3}\right)\text{.}\) This makes sense as \({b}^{-1}=\frac{1}{b}\) for any \(b>1\text{.}\)

What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as \((-\infty ,\infty)\text{.}\)

Look at each graph. What is the range of the function? Notice that the graph is always above and never touches the \(x\)-axis. Thus, the range is all positive numbers. We write the range in interval notation as \(\left(0,\infty \right)\text{.}\)

Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the \(x\)-axis very closely but will never cross it. So we call the line \(y=0\text{,}\) the \(x\)-axis, a horizontal asymptote.

We summarize these results below.

Domain	\((-\infty ,\infty)\)
Range	\(\left(0,\infty \right)\)
\(x\)-intercept	None
\(y\)-intercept	\(\left(0,1\right)\)
Contains	\(\left(1,b\right), \left(-1,\frac{1}{b}\right)\)
Asymptote	\(x\)-axis, the line \(y=0\)

Our definition of an exponential function \(f\left(x\right)={b}^{x}\) says \(b>0\text{,}\) but the examples and discussion so far has been about functions where \(b>1\text{.}\) What happens when \(0<b<1\text{?}\) The next example will explore this possibility.

Example 14.1.11.

On the same coordinate system, graph \(f\left(x\right)={\left(\frac{1}{2}\right)}^{x}\) and \(g\left(x\right)={\left(\frac{1}{3}\right)}^{x}\text{.}\)

Explanation

We will use point plotting to graph the functions.

Checkpoint 14.1.14.

Graph: \(f\left(x\right)={\left(\frac{1}{4}\right)}^{x}\text{.}\)

Explanation

Checkpoint 14.1.16.

Graph: \(g\left(x\right)={\left(\frac{1}{5}\right)}^{x}\text{.}\)

Explanation

Now let's look at the graphs from the previous examples, so we can now identify some of the properties of exponential functions where \(0<b<1\text{.}\)

The graphs of \(f\left(x\right)={\left(\frac{1}{2}\right)}^{x}\) and \(g\left(x\right)={\left(\frac{1}{3}\right)}^{x}\) as well as the graphs of \(f\left(x\right)={\left(\frac{1}{4}\right)}^{x}\) and \(g\left(x\right)={\left(\frac{1}{5}\right)}^{x}\) all have the same basic shape. While this is the shape we expect from an exponential function, the graphs go down from left to right (i.e. they're decreasing), while the previous graphs, when \(b>1\text{,}\) went from up from left to right (i.e. they're increasing).

We notice that for each function, the graph still contains the point (0, 1). This should make sense because \({b}^{0}=1\) for any \(b>0\text{.}\)

As before, the graph of each function, \(f\left(x\right)={b}^{x}\text{,}\) also contains the point \(\left(1,b\right)\text{.}\) The graph of \(f\left(x\right)={\left(\frac{1}{2}\right)}^{x}\) contains \(\left(1,\frac{1}{2}\right)\) and the graph of \(g\left(x\right)={\left(\frac{1}{3}\right)}^{x}\) contains \(\left(1,\frac{1}{3}\right)\text{.}\) This makes sense as \({b}^{1}=b\) for any \(b\text{.}\)

Notice too that the graph of each function, \(f\left(x\right)={b}^{x}\text{,}\) also contains the point \(\left(-1,\frac{1}{b}\right)\text{.}\) The graph of \(f\left(x\right)={\left(\frac{1}{2}\right)}^{x}\) contains \(\left(-1,2\right)\) and the graph of \(g\left(x\right)={\left(\frac{1}{3}\right)}^{x}\) contains \(\left(-1,3\right)\text{.}\) This makes sense as \({b}^{-1}=\frac{1}{b}\) for any \(b>0\text{.}\)

What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as \(\left(-\infty ,\infty \right)\text{.}\) Again, the graph is always above and never touches the \(x\)-axis. Thus, the range is the set of all positive numbers. We write the range in interval notation as \(\left(0,\infty \right)\text{.}\)

We will summarize these properties in Table 14.1.18, which also includes when \(b>1\text{.}\)

Example 14.1.20.

On the same coordinate system, graph \(f\left(x\right)={3}^{x}\) and \(g\left(x\right)={3}^{x}-2\text{.}\)

Explanation

We will use point plotting to graph the functions.

Checkpoint 14.1.23.

On the same coordinate system, graph: \(f\left(x\right)={3}^{x}\) and \(g\left(x\right)={3}^{x}+2\text{.}\)

Explanation

Checkpoint 14.1.25.

On the same coordinate system, graph: \(f\left(x\right)={4}^{x}\) and \(g\left(x\right)={4}^{x}-2\text{.}\)

Explanation

All of our exponential functions so far have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.

Before we can look at this exponential function, we need to define the irrational number, \(e\text{.}\) This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of \({\left(1+\frac{1}{n}\right)}^{n}\) as \(n\) gets larger and larger. We say, as \(n\) approaches infinity, or increases without bound. The table shows the value of \({\left(1+\frac{1}{n}\right)}^{n}\) for several values of \(n\text{.}\)

The number \(e\) is like the number \(\pi \) in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number \(e\) is called the natural base.

Let's graph the function \(f\left(x\right)={e}^{x}\) on the same coordinate system as \(g\left(x\right)={2}^{x}\) and \(h\left(x\right)={3}^{x}\text{.}\)

Notice that the graph of \(f\left(x\right)={e}^{x}\) is between the graphs of \(g\left(x\right)={2}^{x}\) and \(h\left(x\right)={3}^{x}\text{.}\) This should make sense as \(2<e<3\text{.}\)

Subsection 14.1.2 Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function, which is used to calculate compound interest. If you've worked in a laboratory with bacteria, then you are familiar with the exponential growth of bacteria, which can be modeled by an exponential function.