Concept

Reciprocal Function

The reciprocal function is a function that pairs each $x\text{-}$ value with its reciprocal.

$\hspace{1.3cm} x \hspace{1.3cm}$	$\hspace{1.3cm} f(x) \hspace{1.3cm}$
$1$	$1$
$2$	$\frac{1}{2}$
$3$	$\frac{1}{3}$
$4$	$\frac{1}{4}$
$\vdots$	$\vdots$

The function rule of the reciprocal function is obtained algebraically by writing the rule for the pairings shown in the table above.

$f(x) = \dfrac{1}{x}, \quad x\neq0$

The domain and range are all nonzero real numbers. Its graph is a hyperbola, which consists of two symmetrical parts called branches. Furthermore, it has two asymptotes: the $x\text{-}$ and $y\text{-}$ axis.

The reciprocal function is a member of a larger family of functions used to model inverse variations. These functions have the general form $\frac{k}{x},$ where both the constant $k$ and the $x\text{-}$ variable are nonzero. For this reason, the reciprocal function is also known as the inverse variation parent function.

Extra

Reciprocal vs. Inverse of a Function

It is important not to confuse the reciprocal of a function with the inverse of a function. For numbers, $a^{\text{-}1}$ refers to the reciprocal $\frac{1}{a},$ while the notation $f^{\ \text{-} 1}(x)$ is commonly used to refer to the inverse of a function.

Function, $f$	Reciprocal, $\frac{1}{f}$	Inverse, $f^{\text{-}1}$
$f(x) = 2x + 10$	$\dfrac{1}{f(x)}=\dfrac{1}{2x+10}$	$f^{\ \text{-} 1}(x) = \dfrac{1}{2}(x - 10 )$
$f(x) = x^3 - 5$	$\dfrac{1}{f(x)}=\dfrac{1}{x^3-5}$	$f^{\ \text{-} 1}(x) = \sqrt[3]{x + 5 }$