Power Functions and Root Functions

Definition A power function is a function of the form

where $k \ne 0$ is a real constant and the exponent $p$ is a rational constant, often an integer. Irrational exponents such as $x^{\sqrt{2}}$ also give power functions for positive $x$, but defining them needs the exponential functions of a later unit. The domain is the set of real $x$ for which $x^p$ is a real number, and it depends on $p$ as described below. Some books use the name only for $k = 1$.

Definition For a real number $x$ and a positive integer $n$, the power $x^n$ is the product of $n$ factors equal to $x$. For $x \ne 0$ we also set

The expression $0^0$ is left undefined here, and $0^{-n}$ would mean division by zero. So $x^n$ with $n \ge 1$ has domain all real numbers, while $x^0$ and $x^{-n}$ have domain $x \ne 0$. Some formulas, such as the binomial theorem and the power rule of calculus, use the convention that $x^0 = 1$ for every $x$, including $x = 0$, and state it when they need it. When the integer exponent is even the function is even, so its graph is symmetric about the $y$-axis, and its values are positive for $x \ne 0$. When the exponent is odd the function is odd, so its graph is symmetric about the origin, and each value has the sign of $x$.

Definition Let $n \ge 2$ be an integer and $a$ a real number.

• If $n$ is odd, $\sqrt[n]{a}$ is the unique real number $b$ with $b^n = a$.
• If $n$ is even and $a \ge 0$, $\sqrt[n]{a}$ is the unique real number $b \ge 0$ with $b^n = a$.
• If $n$ is even and $a < 0$, no real number $b$ has $b^n = a$, so $\sqrt[n]{a}$ is not a real number.

In the first two cases $\sqrt[n]{a}$ is called the principal $n$th root of $a$. For $n = 2$ we write $\sqrt{a}$.

Definition Let $n \ge 2$ be an integer and consider the root function $f(x) = \sqrt[n]{x}$.

• For even $n$ its domain and range are both $[0, \infty)$.
• For odd $n$ its domain and range are both all real numbers, and it is an odd function, $\sqrt[n]{-x} = -\sqrt[n]{x}$.

In both cases the function is increasing on its domain.

Definition Let $\frac{m}{n}$ be a rational number in lowest terms with integers $m$ and $n \ge 2$, and let $a$ be a real number for which $\sqrt[n]{a}$ is real, with $a \ne 0$ when $m < 0$. Then

For $a > 0$ the usual laws $a^r a^s = a^{r + s}$, $(a^r)^s = a^{rs}$ and $(ab)^r = a^r b^r$ (with $b > 0$ too) hold for all rational $r$ and $s$. For a negative base the exponent must be written in lowest terms, and these laws can fail.

Definition For $f(x) = x^{m/n}$ with $\frac{m}{n}$ in lowest terms, $n \ge 2$ and $m \ne 0$, the natural domain is given by this table.

For a formula built from several pieces, every radicand of an even root must be nonnegative and every denominator must be nonzero.

Definition Let $n \ge 2$ be an integer. If $n$ is odd, $x^n$ is one-to-one on all real numbers and its inverse is $\sqrt[n]{x}$. If $n$ is even, $x^n$ is not one-to-one on all real numbers, since $(-x)^n = x^n$, but its restriction to $[0, \infty)$ is one-to-one with range $[0, \infty)$, and the inverse of that restriction is $\sqrt[n]{x}$. Accordingly $\left(\sqrt[n]{x}\right)^n = x$ for every $x$ in the domain of the root, while $\sqrt[n]{x^n} = x$ holds for every real $x$ when $n$ is odd but only for $x \ge 0$ when $n$ is even. More generally, for a rational $p \ne 0$ the functions $x^p$ and $x^{1/p}$ on $(0, \infty)$ are inverses of each other.

Definition For every real number $x$,

More generally, for an integer $n \ge 2$ the root $\sqrt[n]{x^n}$ equals $\lvert x \rvert$ when $n$ is even and equals $x$ when $n$ is odd. In particular $\sqrt{x^2} = x$ holds exactly when $x \ge 0$.