Monomial and Power Function¶

Definition¶

The function $f(x)=a x^k$ is called a monomial if the exponent $k$ is a natural number or a power function if $k$ is allowed to be a real number. When $k$ is a natural number, we usually call it the degree of the monomial, and the real number $a$ is called the coefficient.

Interpretation¶

Monomials and power functions are the simplest building blocks of algebraic growth: one coefficient times one power of $x$ . The exponent $k$ decides the shape of the graph, while the coefficient $a$ only stretches it vertically or, when $a<0$ $0$ , also reflects it in the $x$ -axis.

Graphs of x squared, x cubed, the square root of x and 1/x, which has two branches since it is undefined at 0.

The figure shows four standard shapes, all passing through $(1,1)$ because $1^k=1$ $1$ . The even power $x^2$ is symmetric about the $y$ -axis and the odd power $x^3$ is symmetric about the origin. The root $x^{1/2}=\sqrt{x}$ is drawn only for $x\ge0$ $0$ , and $x^{-1}$ has two branches because it is not defined at $0$ .

Change the exponent $k$ and the coefficient $a$ in $f(x)=ax^k$ $a x^{k}$ : integer exponents give graphs on all of $\mathbb R$ (except $x=0$ when $k<0$ $0$ ); for general real exponents we plot $x>0$ $0$ . Rational exponents with odd denominator can also be defined for negative $x$ , although this panel does not plot that branch.

The exponent also fixes the domain. For a natural number $k$ , $ax^k$ is defined on all of $\mathbb{R}$ ; an even $k$ gives $f(-x)=f(x)$ and an odd $k$ gives $f(-x)=-f(x)$ $- f (x)$ . A negative integer exponent means $x^{-n}=1/x^n$ $1/ x^{n}$ , so $x=0$ is excluded, while for a general real exponent $x^k$ is defined for $x>0$ $0$ , as described under exponentiation. Comparing exponents depends on where $x$ lies: on $(1,+\infty)$ a larger exponent gives a larger value, but on $(0,1)$ the order is reversed, for example $(\tfrac{1}{4})^3=\tfrac{1}{64}<(\tfrac{1}{4})^2=\tfrac{1}{16}<(\tfrac{1}{4})^{1/2}=\tfrac{1}{2}$ $\frac{1}{2}$ .

A common slip is to confuse a power function with an exponential function. In $x^k$ the variable is the base and the exponent is fixed; in $b^x$ the base is fixed and the variable is the exponent. Monomials are the building blocks of polynomials, which are finite sums of them.

Key Results¶

The coefficient controls scaling, while the exponent controls growth behaviour.
Power functions with a negative exponent have a singularity at $0$ (for example $x^{-1}$ ). For a non-integer exponent $k\text{,}$ $x^k$ is in general defined only for $x>0$ (or $x\geq 0$ when $k>0$ $0$ ).

Examples¶

$f(x)=3x^2$ is a monomial of degree $2\text{.}$
$f(x)=x^{-1}$ is a power function with a singularity at $0\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Aliases: power function Symbols: infinity, real numbers, square root

Last updated: 20 October 2026 03:02