Monomial, and, more generally, Laurent monomial functions are specific examples of a much larger class of functions called power functions, as defined below.

Definition powerfunction broadens our scope of functions to include non-integer exponents such as \(f(x) = 2x^{4/3}\), \(g(t) = t^{0.4}\) and \(h(w) = w^{\sqrt {2}}\). Our primary aim in this section is to ascribe meaning to these quantities.

1 Rational Number Exponents

The road to real number exponents starts by defining rational number exponents.

There are quite a few items worthy of note which are consequences of Definition 2. First off, if \(m\) is an integer, then \(x^{\frac {m}{1}} = x^{m}\) so expressions like \(x^{\frac {3}{1}}\) are synonymous with \(x^3\), as we would expect. Second, the definition of \(x^{\frac {m}{n}}\) can be taken as just \(\left (\sqrt [n]{x}\right )^m\) and shown to be equal to \(\sqrt [n]{x^m}\) (or vice-versa) courtesy of properties of radicals. We state both in Definition 2 to allow for the reader to choose whichever form is more convenient in a given situation. The critical point to remember is no matter which representation you choose, keep in mind the restrictions if \(n\) is even, \(x \geq 0\) and if \(m < 0\), \(x \neq 0\).

Moreover, per this definition, \(x^{\frac {1}{n}} = \sqrt [n]{x^{1}} = \sqrt [n]{x}\), so we may rewrite principal roots as exponents: \(\sqrt {x} = x^{\frac {1}{2}}\) and \(\sqrt [5]{x} = x^{\frac {1}{5}}\). This makes sense from an algebraic standpoint since per Theorem ??, \(\left (\sqrt [n]{x} \right )^n = x\). Hence if we were to assign an exponent notation to \(\sqrt [n]{x}\), say \(\sqrt [n]{x} = x^r\), then \(\left (\sqrt [n]{x}\right )^n = (x^r)^n = x\). If the properties of exponents are to hold, then, necessarily, \( (x^r)^n = x^{rn} = x = x^{1}\), so \(rn = 1\) or \(r = \frac {1}{n}\). While this argument helps motivate the notation, as we shall see shortly, great care must be exercised in applying exponent properties in these cases. The long and short of this is that root functions as defined in Section ?? are all members of the ‘power functions’ family.

Another important item worthy of note in Definition 2 is that it is absolutely essential we express the rational number \(r\) in lowest terms before applying the root-power definition. For example, consider \(x^{0.4}\). Expressing \(r\) in lowest terms, we get: \(r = 0.4 = \frac {4}{10} = \frac {2}{5}\). Hence, \(x^{0.4} = x^{2/5} = (\sqrt [5]{x})^2\) or \(\sqrt [5]{x^2}\), either of which is defined for all real numbers \(x\). In contrast, consider the equivalence \(r = 0.4 = \frac {4}{10}\). Here, the expression \((\sqrt [10]{x})^4\) is defined only for \(x \geq 0\) owing to the presence of the even indexed root, \(\sqrt [10]{x}\). Hence, \((\sqrt [10]{x})^4 \neq x^{\frac {4}{10}} = x^{\frac {2}{5}}\) unless \(x \geq 0\). On the other hand, the expression \(\sqrt [10]{x^4}\) is defined for all numbers, \(x\), since \(x^4 \geq 0\) for all \(x\). In fact, it can be shown that \(\sqrt [10]{x^4} = \sqrt [5]{x^2}\) for all real numbers. This means \(\sqrt [10]{x^4} = \sqrt [5]{x^2} = x^{\frac {2}{5}} = x^{\frac {4}{10}}\). So, to review, in general we have: \(x^{\frac {4}{10}} = \sqrt [10]{x^4}\), but \(x^{\frac {4}{10}} \neq \left (\sqrt [10]{x}\right )^{4}\) unless \(x \geq 0\). Once again the easiest way to avoid confusion here is to reduce the exponent to lowest terms before converting it to root-power notation.

Likewise, we have to be careful about the properties of exponents when it comes to rational exponents. Consider, for instance, the product rule for integer exponents: \(x^{m} x^{n} = x^{m+n}\). Consider \(f(x) = x^{\frac {1}{2}} x^{\frac {1}{2}}\) and \(g(x) = x^{\frac {1}{2} + \frac {1}{2}}\). In the first case, \(f(x) = x^{\frac {1}{2}} x^{\frac {1}{2}} =\sqrt {x} \sqrt {x} = (\sqrt {x})^2 = x\) only for \(x \geq 0\). In the second case, \(g(x) = x^{\frac {1}{2} + \frac {1}{2}} = x^{\frac {2}{2}} = x^{1} = x\) for all real numbers \(x\). Even though \(f(x) = g(x)\) for \(x \geq 0\), \(f\) and \(g\) are different functions since they have different domains.

Similarly, the power rule for integer exponents: \((x^n)^m = x^{nm}\) does not hold in general for rational exponents. To see this, consider the three functions: \(f(x) = (x^{\frac {1}{2}} )^2\), \(g(x) = x^{\frac {2}{2}}\), and \(h(x) = (x^2)^{\frac {1}{2}}\). In the first case, \(f(x) = (x^{\frac {1}{2}})^2 = (\sqrt {x})^2 = x\) for \(x \geq 0\) only (this is the same function \(f\) above.) In the second case, the rational number \(r = \frac {2}{2} = 1\), so \(g(x) = x^{\frac {2}{2}} = x^{\frac {1}{1}} = x^{1} = x\) for all real numbers, \(x\) (this is the same function \(g\) from above.) In the last case, \(h(x) = (x^2)^{\frac {1}{2}} = \sqrt {x^2} = |x|\) for all real numbers, \(x\). Once again, despite \(f(x) = g(x) = h(x)\) for all \(x \geq 0\), \(f\), \(g\) and \(h\) and are three different functions. We graph \(f\), \(g\), and \(h\) below.

In general, the properties of integer exponents do not extend to rational exponents unless the bases involved represent non-negative real numbers or the roots involved are odd. We have the following:

Next, we turn our attention to the graphs of \(f(x) =x^r = x^{\frac {m}{n}}\) for varying values of \(m\) and \(n\). When \(n\) is even, the domain is restricted owing to the presence of the even indexed root to \([0, \infty )\). The range is likewise \([0, \infty )\), a fact leave to the reader. All of the functions below are increasing on their domains, and it turns out this is always the case provided \(r>0\). There is, however, is a difference in how the functions are increasing - and this is the concept of concavity. As with many concepts we’ve encountered so far in the text, concavity is most precisely defined using Calculus terminology, but we can nevertheless get a sense of concavity geometrically. For us, a curve is concave up over an interval if it resembles a portion of a ‘\(\smile \)’ shape. Similarly, a curve is called concave down over an interval if resembles part of a ‘\(\frown \)’ shape. When \(0 < r < 1\), the graphs of \(f(x) = x^r\) resemble the left half of \(\frown \) and so are concave down; when \(r>1\), the graphs resemble the right half of a ‘\(\smile \)’ and are hence described as ‘concave up.’

Below we graph several examples of \(f(x) =x^r = x^{\frac {m}{n}}\) where \(n\) is odd. Here, the domain is \((-\infty , \infty )\) since the index on the root here is odd. Note that when \(m\) is even, the graphs appear to be symmetric about the \(y\)-axis and the range looks to be \([0, \infty )\). When \(m\) is odd, the graphs appear to be symmetric about the origin with range \((-\infty , \infty )\). We leave verification of these facts to the reader. Note here also that for \(x \geq 0\), the graphs are down for \(0<r<1\) and concave up for \(r > 1\).

When \(r<0\), we have variables appear in the denominator which open the opportunities for vertical and horizontal asymptotes. Below are graphed two examples

Unsurprisingly, Theorem ??, which, as stated, applied to root functions, generalizes to all rational powers.

The proof of Theorem 2 is identical to that of Theorem ??, and we suggest the reader work through the details. We give Theorem 2 a test run in the following example.

We now turn our attention to more complicated functions involving rational exponents.

2 Real Number Exponents

We wish now to extend the concept of ‘exponent’ from rational to all real numbers which means we need to discuss how to interpret an irrational exponent. Once again, the notions presented here are best discussed using the language of Calculus or Analysis, but we nevertheless do what we can with the notions we have.

Consider the wildly famous irrational number \(\pi \). The number \(\pi \) is defined geometrically as the ratio of the circumference of a circle to that circle’s diameter. The reason we use the symbol ‘\(\pi \)’ instead of any numerical expression is that \(\pi \) is an irrational number, and, as such, its decimal representation neither terminates nor repeats. Hence we approximate \(\pi \) as \(\pi \approx 3.14\) or \(\pi \approx 3.14159265\). No matter how many digits we write, however, what we have is a rational number approximation of \(\pi \).

The good news is we can approximate \(\pi \) to any desired accuracy using rational numbers by taking enough digits, so while we’ll never ‘reach’ the exact value of \(\pi \) with rational numbers, we can get as close as we like to \(\pi \) using rational numbers. That being said, we assume \(\pi \) exists on the real number line, despite the fact the list of digits to pinpoint its location is, in some sense, infinite.

We take this tack when defining the value of a number raised to an irrational exponent. Consider, for instance, \(2^{\pi }\). We can compute \(2^3 = 8\), \(2^{3.1} = 2^{\frac {31}{10}} = \sqrt [10]{2^{31}} \approx 8.574 \), \(2^{3.14} = 2^{\frac {314}{100}} = 2^{\frac {157}{50}} = \sqrt [50]{2^{157}} \approx 8.8512\), and so on, so one way to define \(2^{\pi }\) as the unique real number we obtain as the exponents ‘approach’ \(\pi \).

It is with this understanding that we present the notion of a ‘power function,’ as described in Definition 1: \(f(x) = a x^p\) where \(a\) and \(p\) are nonzero real number parameters. Here the exponent \(p\) is open to any (nonzero) real number. Because of how we define real number exponents, if \(p\) is irrational, then \( x \geq 0\) to avoid having negatives under even-indexed roots as we go through the approximation process.

In general, real number exponents inherit their properties from rational number exponents. For instance, Theorem 1 also holds for all real number exponents and the graphs of power functions inherit their behavior from graphs of rational exponent functions. More specifically, the graphs of functions of the form \(f(x)= x^p\) where \(p>0\) all contain the points \((0,0)\) and \((1,1)\). Moreover, these functions are increasing and their graphs are concave down if \(0<p<1\) and concave up if \(p>1\).

Theorem 2 generalizes to real number power functions, so, for instance to graph \(F(x) = (x-2)^{\pi }\), one need only start with \(y = x^{\pi }\) and shift horizontally two units to the right. (See the Exercises.)

We close this section with an application to economics. According to the US Census, Table 2, the share of money income (2014-2015) is given in the table below on the left. From these data, we can create a cumulative distribution, \(y = L(x)\) called the Lorenz Curve.

The number \(L(x)\) gives the percentage of the total national income earned by the bottom \(x\) percent of wage earners, ranked from lowest income to highest income. Since the population here is separated into ‘quintiles,’ each data point corresponds to \(20 \%\) of the population. So, for example, \(L(20)\) is the percentage of money income earned by the lowest \(20 \%\) of wage earners. In this case, we see \(L(20) = 3.1\). The number \(L(40)\) is the percentage of the money income earned by the bottom \(40 \%\) of wage earners - so this includes not only the money from the Second Quintile, but also the Lowest Quintile: \(L(40) = 8.2 + L(20) = 8.2 + 3.1 = 11.3\). Likewise, \(L(60)\) is the total income share of the bottom \(60 \%\) of wage earners which includes the income from the Middle, Second, and Lowest Quintiles: \(L(60) = 14.3 + L(40) = 14.3 + (8.2+3.1) = 25.6\).

Continuing in this manner, we get \(L(80) = 48.8\) and \(L(100) = 100\), which is what we would expect: \(100 \%\) of the income is earned by \(100 \%\) of the population. We summarize these findings below on the right.

\( \begin{array}{cc} \text {Portion of Population} & \text {Percent of Money Income} \\ \text {Lowest Quintile} & \text { 3.1} \\ \text {Second Quintile} &\text { 8.2} \\ \text {Middle Quintile} & \text {14.3} \\ \text {Fourth Quintile} & \text { 23.2} \\ \text {Highest Quintile} & \text {51.2 } \\ \end{array} \) \( \begin{array}{cc} \text {percent wage earners, $x$} & \text {percent income, $L(x)$} \\ \text { 20} & \text {3.1}\\ \text {40} & \text {11.3} \\ \text {60} & \text {25.6} \\ \text {80 }& \text {48.8} \\ \text {100} & \text {100} \\ \end{array} \)