Expressing Functions as Power Functions

A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers. Polynomials are made of power functions. Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents). The domain of a power function can sometimes be all real numbers, but generally a non-negative value is used to avoid problems with simplifying. The domain of definition is determined by each individual case. Power functions are a special case of power law relationships, which appear throughout mathematics and statistics.

The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series: 

$\displaystyle{\sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n}}$

where $n!$ denotes the factorial of $n$ and $f^na$ denotes the $n$th derivative of $f$ evaluated at the point $x=a$. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. If the Taylor series is centered at zero, then that series is also called a Maclaurin series:

$\displaystyle{f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n! } \, x^{n}}$

Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions ($x^n$) of integer exponent.

$\sin x$ in Taylor Approximations

Figure shows $\sin x$ and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. As more power functions with larger exponents are added, the Taylor polynomial approaches the correct function.

Examples

Functions of the form $f(x) = x^3$, $f(x) = x^{1.2}$, $f(x) = x^{-4}$ are all power functions.