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Reciprocal Function

Concept

Reciprocal Function

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

x\hspace{1.3cm} x \hspace{1.3cm} f(x)\hspace{1.3cm} f(x) \hspace{1.3cm}
11 11
22 12\frac{1}{2}
33 13\frac{1}{3}
44 14\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)=1x,x0f(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-x\text{-} and y-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 kx,\frac{k}{x}, where both the constant kk and the x-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-1a^{\text{-}1} refers to the reciprocal 1a,\frac{1}{a}, while the notation f -1(x)f^{\ \text{-} 1}(x) is commonly used to refer to the inverse of a function.

Function, ff Reciprocal, 1f\frac{1}{f} Inverse, f-1f^{\text{-}1}
f(x)=2x+10f(x) = 2x + 10 1f(x)=12x+10\dfrac{1}{f(x)}=\dfrac{1}{2x+10} f -1(x)=12(x10)f^{\ \text{-} 1}(x) = \dfrac{1}{2}(x - 10 )
f(x)=x35f(x) = x^3 - 5 1f(x)=1x35\dfrac{1}{f(x)}=\dfrac{1}{x^3-5} f -1(x)=x+53f^{\ \text{-} 1}(x) = \sqrt[3]{x + 5 }