Power Functions

Some topics in mathematics are absolutely crucial, not only because of their applications or specific mathematical meaning, but because they are covering a manifold basic tools without which it is nearly impossible to study any other topics at all. This is similar to such elementary knowledge as alphabet when you are studying language or periodic table when you are studying chemistry. Power functions, which we will discuss here, are just one of those indispensable tools in the studying of functions and processes they describe. You will meet power functions in real and complex analysis as the building blocks from which any function could be constructed. You will meet them in physics where there are hundreds of processes which could be described exactly or asymptotically with the help of power functions. In higher algebra such objects as monomials and polynomials are precisely power functions with natural powers and their sums and products. In programming these type of functions is an absolute necessity when working with objective functions or when evaluating the complexity of various algorithms.

What is a power function?

Let's say you have constructed a square out of four line segments of the length $\ell$ (see picture below). How to find its area $S$? Well, this is an easy question: you just need to find the product $AB\cdot BC$. Therefore, $S = \ell\cdot \ell = \ell^{2}$.

Now imagine that you have constructed a cube with the edge of the same length $\ell$:

Let's find its volume. It is the product of the lengths of its edges that meet in the same vertex. Therefore, it equals to $\ell^{3}$. Note that if you double the length of a segment $\ell$, then the area of the square will become four times larger, and the volume of the cube will increase $8$ times. Such growths are called quadratic and cubic, respectively. Another example: if you increase the sides of a cubic box by $3$ times, then it will fit $27$ times more content.

In the same way, if the area of a square is $S$, then the length of its side is $\sqrt{S}$. And if the volume of a cube is $V$, then the length of its edge is $\sqrt[3]{V}$. Keep in mind that $\sqrt{a} = a^{\frac{1}{2}}$and $\sqrt[3]{a} = a^{\frac{1}{3}}$, therefore, you can write down all the mentioned lengths, areas, and volumes in terms of powers $(a^{\frac{1}{3}}, a^{\frac{1}{2}},a^{2},a^{3})$. To work with such entities, we use power functions.

These functions find an important application in estimation of complexity of algorithms. The idea here is simple: for given amount of input data, what is the amount of computational resources that we have to spend in the process of algorithm execution. Quite often, the estimate of the complexity of an algorithm turns out to be a power function. For example, the complexity of the bubble sort algorithm of a list of length $n$ is estimated with the $n^2$ function, which means that the time spent on sorting is on average proportional to the square of the list's length (e.g. if the length of a list, which is being bubble sorted is doubled, the time of sorting would increase in 4 times).

Definition

You can define a function as a power function if it is of the form $f(x) = ax^{p}$ (here $a\ne 0$, and $p$ is any real number). Power functions have extremely varied behavior, depending on the value of parameters $p$ and $a$. Furthermore, the domains and codomains of these functions are crucially sensitive to parameter changes.

Natural powers

To begin with, let's consider only natural values of $p$ ($p = 1,2,3,\dots$) and set $a = 1$. Now the function takes $x$ as an input and returns $\underbrace{x\cdot x\dots x}_{p \text{\ times}}$. You can apply this operation to any real number, that's why when $p$ is natural the domain of power function is the all real axis. However, some subtleties arise when talking about codomain. Note that if $x$ is negative, then $x, x^{3}, x^{5} \dots$ are also negative, but $x^{2}, x^{4}, x^{6}, \dots$ are positive. In general, if $p$ is odd,

$(- x)^{p} = - x^{p}$Then, if $p$ is even,

$(- x)^{p} = x^{p}.$

These properties of power functions with natural powers allow us to draw conclusions about what their graphs look like. For odd functions graphs will be symmetrical with respect to the coordinate center (see examples in the following pictures):

Similarly, the plots of $ax^{p}$ for even $p$ are symmetrical with respect to $y$-axis:

To have the full picture, let's discuss some other properties of $ax^{p}$. First of all, for any natural $p$, power function $ax^{p}$ is continuous in its domain. Showing this, however, goes beyond the scope of this topic. Since for every even $p$ the expression $x^{p} \geq 0$ for any $x$, power functions in these cases are bounded below by zero if $a > 0$, and bounded above by zero if $a < 0$. In the case of an odd $p$, the power function is unbounded and increases monotonically if $a > 0$ and decreases monotonically if $a < 0$.

Let's now extend this class of functions to negative powers.

Integer powers

Before defining power functions for negative powers, let's remember that $x^{- n} = 1 / x^{n}$. And here you would have an obvious problem, because if $x = 0$, it would be impossible to define this fraction. Therefore, if $p < 0$ and is an integer, the domain of $ax^{p}$ is $(- \infty;0)\cup(0;+\infty)$. One more important point is that for natural $n$, if $x^{n}$ is increasing on a line segment, then $x^{- n}$ is decreasing on the same segment.

Otherwise, for positive and negative powers, the functions are similar in many ways, both are continuous over the entire domain: they are even in the case of even powers, and odd for odd ones. You can see the examples of these functions in the following pictures.

As you can see, it is a 'well-behaved' function almost everywhere, but in $x = 0$ the function diverges, meaning that the closer $x$ is to $0$, the greater the absolute value of the function, and this growth is unlimited.

Rational and real powers

Now let's think about power functions $ax^{p}$ that have real $p$, but firstly, let's conduct a little experiment with numbers, namely let's try to raise a non-positive number to a fractional power. For example,

$(- 8)^{\frac{1}{3}} = \sqrt[3]{- 8}.$By definition, $\sqrt[3]{x}$ is such a number that its cube is $x$, therefore, it is quite natural to think that $\sqrt[3]{- 8 } = - 2$. Let's now consider the following exponent:

$(- 8)^{\frac{2}{6}} = \sqrt[6]{(- 8)^{2}} = \sqrt[6]{64} = 2.$And here you would face a crucial problem, because $2/6 = 1/3$, entailing $(- 8)^{\frac{2}{6}} = (- 8)^{\frac{1}{3}}$, but $2 \ne -2$. You could solve issues like this with the help of complex analysis or accurate definitions, however, generally, in real analysis it is common to use exponentiation only for strictly positive bases.

If you are dealing with the function $ax^{p}$, where $p$ is some arbitrary real number, you suppose that the domain of this function is $(0; +\infty)$ or $[0;+\infty)$, where the value of the function is zero when $p$ is positive, or is not defined if $p < 0$. In this domain, the power function is continuous and monotonous. Its behavior is very similar to the above-mentioned functions. The sign of $a$ defines the increase or decrease of function, the change of this sign flips the function symmetrically over $x$-axis. You can find some examples in the following pictures.

Example

In the picture below, there is a cylinder with the height $h$, which is two times bigger than its radius $R$. You can find the volume of the cylinder using the formula $V = Sh$, where $S$ is the area of the disk in the base of the cylinder. Therefore, in this case, you can find the volume of the cylinder as function of $R$. Since the area of the disk is $S = \pi R^2$, the volume is $V = V(R) = S\cdot h = \left(\pi R^2\right)\cdot (2R)$. Simplifying this formula, we obtain the following one:$\boxed{V(R) = 2\pi R^{3}}.$

Conclusion

Below are the most important points to remember from this topic.

• Power function has the form of $f(x) = ax^{p}$, where $a\ne 0$ and $p$ is any real number.

• If $p$ has an odd natural value, $(- x)^{p} = - x^{p}$, and if $p$ has an even natural value, $(- x)^{p} = x^{p}$.

• For a function with odd power $p$, the graph will be symmetrical with respect to the coordinate center. For even $p$, the graph of the power function will be symmetrical with respect to $y$-axis.

• For any positive integer $p$, power function $f(x) = ax^{-p} = a \cdot \frac {1}{ x^{p}}$. In this case, we can't define it if $x = 0$.

• If the power function $ax^n$ increases on a line segment for natural $n$, then $ax^{- n}$ decreases on the same segment.

• Similarly, if the power function $ax^n$ decreases on a line segment for natural $n$, then $ax^{- n}$ increases on the same segment.

• If $p$ is an arbitrary positive real number, then the domain of the power function $f(x) = ax^{p}$ is $[0,+\infty)$. If $p$ is an arbitrary negative real number, then the domain of the power function $f(x) = ax^{p}$ is $(0,+\infty)$.

• For any real values of $p$ and $a$ (while they are both nonzero), all the values of the power function $f(x) = ax^{p}$ have the same sign (all are negative or all are positive) on the segment $(0,+\infty)$.

• The power function $ax^{p}$ is monotonic in its whole domain: for $a> 0$ and $p>0$ it increases, for $a > 0$ and $p < 0$ it decreases, for $a < 0$ and $p > 0$ it decreases, for $a<0$ and $p < 0$ it increases.