A power function has the form

$$f(x)=x^a,$$

where the exponent $a$ is fixed. Different kinds of exponents produce familiar families of functions.

Integer exponents

Positive integer exponents give polynomial behavior. For example,

$$x^2,\qquad x^3$$

produce the basic square and cube functions.

Negative integer exponents give reciprocal powers:

$$x^{-n}=\frac1{x^n},$$

so $x=0$ is excluded from their domains.

Rational exponents and roots

A rational exponent connects powers with roots:

$$x^{1/n}=\sqrt[n]{x},$$

and more generally

$$x^{m/n}=\sqrt[n]{x^m}$$

where the real expression is defined.

The domain depends on the root involved. For example, $\sqrt{x}$ requires $x\ge0$, while $\sqrt[3]{x}$ is defined for every real $x$.

Shape and symmetry

The exponent controls both the graph and its symmetry. Even powers such as $x^2$ give the same output for $x$ and $-x$, while odd powers such as $x^3$ change sign. Reciprocal and root powers introduce different domain and asymptotic behavior.

Power functions provide a common language for polynomial terms, reciprocals and roots.