Power functions

Functions: Power functions

Power functions

Let $a=\frac{p}{q}$ be a rational number that is not an integer, with integers $p$ and $q$ unequal to zero and $q\gt0$ . Then $x^a$ is a function of $x$ with domain $\ivco{0}{\infty}$ if $a\gt0$ and $\ivoo{0}{\infty}$ if $a\lt0$ . After all, $x^a = \sqrt[q]{x^p}$ is the unique number $b\ge0$ for which $b^q =x^p$ applies.

Even if $a$ is not rational, we can define $x^a$ on $\ivco{0}{\infty}$ if $a\gt0$ and on $\ivoo{0}{\infty}$ if $a\lt0$ . Through numbers of the form $\sqrt[q]{x^p}$ , with $\frac{p}{q}$ a rational number close to $a$ , the number can be approximated as well as needed. Technically, this is a bit more difficult than the rational case.

Power Function

Let $a$ be a real number. The function $x^a$ of $x$ is called the power function with exponent $a$ .

If $a=0$ , then $x^a$ is the constant function, although it is not defined in $0$ .

The domain can be chosen in the same way as in the case in which $a$ is rational.

If $a\gt0$ then it could still be useful to assume $0^a =0$ , and hence, expanding the domain of the power function to $\ivco{0}{\infty}$ . But if we do not want to endanger the rules for powers stated below, we keep the domain as $\ivoo{0}{\infty}$ .

All known rules for fractional powers also apply here.

Rules of calculation for powers

Let $a$ and $b$ be real numbers and $x$ and $y$ positive numbers. Then, the following equalities apply.

$\left(x^a\right)^b = x^{a\cdot b}$
$\left(x\cdot y\right)^a = x^a\cdot y ^a$
$x^a\cdot x^b = x^{a+b}$
$\frac{x^a}{x^b}=x^{a-b}$
$x^0 = 1$

These rules lead to the following useful properties of power functions. Here we consider $x^a$ as a function with domain $\ivoo{0}{\infty}$ . If $a$ is a non-zero, then the function $\mathbb{R}\setminus\{0\}$ (all numbers except $0$ ) is defined, but then rules as $x^3 = x^{\frac{6}{2}} =\sqrt[2]{x^{6}}$ no longer apply because $(-1)^3=-1$ and $\sqrt[2]{(-1)^{6}}=1$ .

Let $a$ be a real number unequal to $0$ . The power function $x^a$ with domain $\ivoo{0}{\infty}$

has range $\ivoo{0}{\infty}$ ,
is increasing if $a\gt 0$ and decreasing if $a\lt0$ .

Exclusion $x=0$ is necessary because, for example $x^{-1}$ now occurs and is equal to $\frac{1}{x}$ .

Exclusion $a=0$ is necessary because $x^0$ is constant function $1$ , and hence, only has range $1$ .

Which function corresponds with the graph below?

The function $x^{-3}$ , or $\dfrac{1}{x^{3}}$ corresponds with the graph.

After all, $x^{-3}$ is a function that takes on ever increasing values of $y$ for values of $x$ close to $0$ . For values of $x$ bigger than $1$ , the values of $y$ become increasingly smaller.
Moreover for $x=2$ holds $2^{-3}={{1}\over{8}}$ . This corresponds with the graph.

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