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. The real number $a$ is the coefficient. For a nonzero monomial ($a\ne$$a\ne0$$0$), $k$ is its degree. The zero function ($a=$$a=0$$0$) has no degree under this convention, since every natural exponent gives the same zero monomial.

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.

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 $0$.

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¶

• Function
• Exponentiation and Power

Related Concepts¶

• Exponential Function
• $n$-th Root

Sources¶

• Math I: Calculus · Lecture 4

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

Last updated: 27 October 2026 11:52