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