The reciprocal function is a function that pairs each $x\text{-}$value with its reciprocal. The function rule of this reciprocal function is obtained algebraically by writing the rule for the pairings shown in the table.

The graph of the function $f(x)=\frac{1}{x}$ is a hyperbola, which consists of two symmetrical parts called branches. It has two asymptotes, the $x\text{-}$ and $y\text{-}$axes. The domain and range are all nonzero real numbers.

The graph of $y=\frac{1}{x}$ can be used to graph other reciprocal functions. This can be done by applying different transformations.

Name	Equation	Characteristics
Parent Reciprocal Function	$y=\frac{1}{x}$	$$\begin{array}{rl}\text{Domain:}& \mathbb{R}-\{0\}\\ \text{Range:}& \mathbb{R}-\{0\}\\ \text{Asymptotes:}& x\text{- and }y\text{-axes}\end{array}$$
Inverse Variation Functions	$y=\frac{a}{x}$
General Form of Reciprocal Functions	$y=\frac{a}{x-h}+k$	$$\begin{array}{rl}\text{Domain:}& \mathbb{R}-\{h\}\\ \text{Range:}& \mathbb{R}-\{k\}\\ \text{Asymptotes:}& x=h\text{ and }y=k\end{array}$$

The general form of reciprocal functions is also known as the simple rational function.