Of all of the functions we study in this text, exponential functions are possibly the ones which impact everyday life the most. This section introduces us to these functions while the rest of the chapter will more thoroughly explore their properties.

Up to this point, we have dealt with functions which involve terms like \(x^3\), \(x^{\frac {3}{2}}\), or \(x^{\pi }\) - in other words, terms of the form \(x^{p}\) where the base of the term, \(x\), varies but the exponent of each term, \(p\), remains constant.

In this chapter, we study functions of the form \(f(x) = b^{x}\) where the base \(b\) is a constant and the exponent \(x\) is the variable. We start our exploration of these functions with the time-honored classic, \(f(x) = 2^{x}\).

We make a table of function values and plot the corresponding points on a graph furnished by desmos below.

\[ \begin{array}{|r||r|r|} \hline x & f(x) & (x,f(x)) \\ \hline -3 & 2^{-3} = \frac {1}{8} & \left (-3, \frac {1}{8} \right ) \\ \hline -2 & 2^{-2} = \frac {1}{4} & \left (-2, \frac {1}{4} \right ) \\ \hline -1 & 2^{-1} = \frac {1}{2} & \left (-1, \frac {1}{2} \right ) \\ \hline 0 & 2^{0} = 1 & ( 0 ,1) \\ \hline 1 & 2^{1} = 2 & ( 1, 2) \\ \hline 2 & 2^{2} = 4 & (2,4) \\ \hline 3 & 2^{3} = 8 & (3, 8) \\ \hline \end{array} \]

A few remarks about the graph of \(f(x) = 2^{x}\) are in order. As \(x \rightarrow -\infty \) and takes on values like \(x = -100\) or \(x=-1000\), the function \(f(x) = 2^{x}\) takes on values like \(f(-100) = 2^{-100} = \frac {1}{2^{100}}\) or \(f(-1000) = 2^{-1000} = \frac {1}{2^{1000}}\).

In other words, as \(x \rightarrow -\infty \), \(2^{x} \approx \frac {1}{\text {very big $(+)$}} \approx \text {very small $(+)$}\) That is, as \(x \rightarrow -\infty \), \(2^{x} \rightarrow 0^{+}\), so \(\lim \limits _{x \rightarrow -\infty } 2^{x} =0\) . This produces the \(x\)-axis, \(y = 0\) as a horizontal asymptote to the graph as \(x \rightarrow -\infty \). We invite the reader to see this behavior graphically by scrolling towards the left on the Desmos graph above.

On the flip side, as \(x \rightarrow \infty \), we find \(f(100) = 2^{100}\), \(f(1000) = 2^{1000}\), and so on, thus \(\lim _{x \rightarrow \infty } 2^{x} = \infty \). We can see this graphically by scrolling towards the right (and up!) on the Desmos graph above.

Note that the smooth connected appearance of the graph of \(f(x) = 2^{x}\) suggests that not only is \(f\) defined for all real numbers, but also that \(f\) is continuous. Moreover, we are assuming \(f(x) = 2^{x}\) is increasing: that is, if \(a<b\), then \(2^{a} < 2^{b}\). While these facts may appear obvious, the proofs of these properties are best left to Calculus. For us, we assume these properties in order to state the domain of \(f\) is \((-\infty , \infty )\) and the range of \(f\) is \((0, \infty )\). Moreover, since \(f\) is increasing, \(f\) is one-to-one, hence invertible.

Suppose we wish to study the family of functions \(f(x) = b^{x}\). Which bases \(b\) make sense to study? We find that we run into difficulty if \(b < 0\). For example, if \(b = -2\), then the function \(f(x) = (-2)^{x}\) has trouble, for instance, at \(x = \frac {1}{2}\) since \((-2)^{1/2} = \sqrt {-2}\) is not a real number. In general, if \(x\) is any rational number with an even denominator, then \((-2)^{x}\) is not defined, so we must restrict our attention to bases \(b \geq 0\).

What about \(b = 0\)? The function \(f(x) = 0^{x}\) is undefined for \(x \leq 0\) because we cannot divide by \(0\) and \(0^{0}\) is an indeterminate form. For \(x > 0\), \(0^{x} = 0\) so the function \(f(x) = 0^{x}\) is the same as the function \(f(x) = 0\), \(x > 0\). Since we know everything about this function, we ignore this case.

The only other base we exclude is \(b=1\), since the function \(f(x) = 1^{x} = 1\) for all real numbers \(x\), since, once again, a function we have already studied. We are now ready for our definition of exponential functions.

We leave it to the reader to verify that if \(b > 1\), then the exponential function \(f(x) = b^{x}\) will share the same basic shape and characteristics as \(f(x) = 2^{x}\).

What if \(0 < b < 1\)? Consider \(g(x) = \left (\frac {1}{2}\right )^{x}\). We could certainly build a table of values and connect the points, or we could take a step back and note that \(g(x) = \left (\frac {1}{2}\right )^{x} = \left (2^{-1}\right )^{x} = 2^{-x} = f(-x)\), where \(f(x) = 2^{x}\). Per Section Transformations, the graph of \(f(-x)\) is obtained from the graph of \(f(x)\) by reflecting it across the \(y\)-axis as seen in the graph below.

We see that the domain and range of \(g\) match that of \(f\), namely \((-\infty , \infty )\) and \((0,\infty )\), respectively. Like \(f\), \(g\) is also one-to-one. Whereas \(f\) is always increasing, \(g\) is always decreasing. As a result, as \(\lim _{x \rightarrow -\infty } g(x) = \infty \), and on the flip side, \(\lim _{x \rightarrow \infty } g(x) = 0\). (More specifically, \(x \rightarrow \infty \), \(g(x) \rightarrow 0^{+}\).) It shouldn’t be too surprising that for all choices of the base \(0 < b < 1\), the graph of \(y=b^{x}\) behaves similarly to the graph of \(g\).

We summarize the basic properties of exponential functions in the following theorem.

Exponential functions also inherit the basic properties of exponents from Theorem exponentprops. We formalize these below and use them as needed in the coming examples.

In addition to base \(2\) which is important to computer scientists, two other bases are used more often than not in scientific and economic circles. The first is base \(10\). Base \(10\) is called the ‘common base’ and is important in the study of intensity (sound intensity, earthquake intensity, acidity, etc.)

The second base is an irrational number, \(e\). Like \(\sqrt {2}\) or \(\pi \), the decimal expansion of \(e\) neither terminates nor repeats, so we represent this number by the letter ‘\(e\).’ A decimal approximation of \(e\) is \(e \approx 2.718\), so the function \(f(x) = e^{x}\) is an increasing exponential function.

The number \(e\) is called the ‘natural base’ for lots of reasons, one of which is that it ‘naturally’ arises in the study of growth functions in Calculus. We will more formally discuss the origins of \(e\) in Section ExpLogApplications.

It is time for an example.

Our next example showcases an important application of exponential functions: economic depreciation.

Some remarks about Example cardepreciationex are in order. First the function in the previous example is called a ‘decay curve’. Increasing exponential functions are used to model ‘growth curves’ and we shall see several different examples of those in Section ExpLogApplications.

Second, as seen in numbers georatio1 and georatio2, \(V(t+1) = 0.8 V(t)\). That is to say, the function \(V\) has a constant unit multiplier, in this case, \(0.8\) because to obtain the function value \(V(t+1)\), we multiply the function value \(V(t)\) by \(b\). It is not coincidence that the multiplier here is the base of the exponential, \(0.8\).

Indeed, exponential functions of the form \(f(x) = a \cdot b^{x}\) have a constant unit multiplier, \(b\). To see this, note

\[ \frac {f(x+1)}{f(x)} = \frac {a \cdot b^{x+1}}{ a \cdot b^{x}} = b^{1} = b.\]

Hence \(f(x+1) = f(x) \cdot b\). This will prove useful to us in Section ExpLogApplications when making decisions about whether or not a data set represents exponential growth or decay.

More generally, one can show (see Exercise exponentialchangeexercise) for any real number \(x_{0}\) that \(f(x_{0}+\Delta x) = f(x_{0}) b^{\Delta x}\). That is, to obtain \(f(x_{0} + \Delta x)\) from \(f(x_{0})\), we multiply by \(\Delta x\) factors of the constant unit multiplier, \(b\). This is at the heart of what it means to be an exponential function.

If this discussion seems familiar, it should. For linear functions, \(f(x) = mx +b\), we can obtain the slope \(m\) by computing \(f(x+1) - f(x)\). To see this, note \(f(x+1) - f(x) = (m(x+1) +b) - (mx+b) = m\) so that \(f(x+1) = f(x) + m\). In this way, we see that the slope \(m\) is the constant unit addend in that in order to obtain \(f(x+1)\), we add \(m\) to the function value \(f(x)\).

This notion is solidified in the point-slope form of a linear function, Equation linearfunctionpointslope. For for any real numbers \(x\) and \(x_{0}\), we have \(f(x) = f(x_{0}) + m(x-x_{0})\). If we let \(x = x_{0}+ \Delta x\), we get \(f(x_{0}+ \Delta x) = f(x_{0}) + m \Delta x\). In other words, to obtain \(f(x_{0}+\Delta x)\) from \(f(x_{0})\), we add \(m\) times \(\Delta x\).

Taking inspiration from linear functions, we define the ‘point-base’ form of an exponential function below.

Just as the point-slope form of a linear function is helpful in building linear models, the point-base form of an exponential function will prove useful in building exponential models.

Next, while we saw in Example cardepreciationex number cararcex, exponential functions, unlike linear functions, do not have a constant rate of change. However, in numbers carrelarcex1 and carrelarcex2, we see that in some cases, they do have a constant relative rate of change. We define this notion below.

For exponential functions of the form \(f(x) = a \cdot b^{x}\), we compute the relative rate of change over the interval \([x, x+1]\) and find it is constant:

\[ \frac {f(x+1) - f(x)}{f(x)} = \frac {f(x+1)}{f(x)} - \frac {f(x)}{f(x)} = b -1,\]

where we are using the fact that \(\frac {f(x+1)}{f(x)} = b\).

One way to interpret this result is when comparing \(f(x)\) to \(f(x+1)\), the exponential function grows (if \(b>1\)) or decays (if \(b<1\)) by \((b-1) \cdot 100 \%\). In our example, \(V(t) = 25 (0.8)^{t}\) so \(b = 0.8\) and, as we saw, the relative rate of change from \(V(t)\) to \(V(t+1)\) was \( 0.8 - 1= -0.2\), meaning the value of the car over the course of one year depreciates by \(20 \%\).

We close this section with another important application of exponential functions, Newton’s Law of Cooling.