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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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*.

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, $f^{\text{-}1}$ |
---|---|---|

$f(x) = 2x + 10$ | $\dfrac{1}{f(x)}=\dfrac{1}{2x+10}$ | $f^{\ \text{-} 1}(x) = \dfrac{1}{2}(x - 10 )$ |

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