Tool to calculate the reciprocal of a function f, i.e. the inverse function f-1 which applied to the first function returns the initial value x.

Reciprocal Function - dCode

Tag(s) : Functions

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The reciprocal of a function $ f $ is written $ f^{(-1)} $ is such that the following equation is true: $$ f^{(-1)}(f(x)) = x $$

That is to say that it is the mathematical function which *cancels* the effects of another function.

__Example:__ The reciprocal of the exponential function $ \exp(x) $ is the natural logarithm function $ \ln(x) $ because $ \exp( \ln (x) ) = x $

Although the reciprocal function is denoted with $ ^{-1} $ as the inverse $ 1/x $ function, be careful not to confuse the two.

To find the expression of the inverse of a function $ f(x) $, express $ x $ as a function of $ f(x) $ (to facilitate calculations, write $ f(x) = y $ and express $ f^{(-1)}(y) $)

__Example:__ To calculate the reciprocal of $ f(x) = y = 2x $, it is to calculate $ x = y/2 $ therefore the reciprocal of $ f^{(-1)}(y) = y/2 $ which checks $ f^{(-1)}(f(x)) = (2x)/2 = x $

Here are some of the most common reciprocal functions:

Function $ f(x) $ | Inverse $ f^{(-1)}(x) $ |
---|---|

$ x + a $ | $ x − a $ |

$ k.x $ | $ x/k $ |

$ x^2 $ | $ \sqrt{x} $ |

$ x^k $ | $ \sqrt[k]{x} $ |

$ \exp(x) $ | $ \ln(x) $ |

$ a^x $ | $ \log_a(x) $ |

$ \sin(x) $ | $ \arcsin(x) $ |

$ \cos(x) $ | $ \arccos(x) $ |

$ \tan(x) $ | $ \arctan(x) $ |

Yes, it has been shown that if the reciprocal of a function exists then there is only one, it is unique.

The 1/x inverse function $ f(x) = 1/x $ is its own reciprocal function, it is said to be involutive.

__Example:__ $ f(1/x) = 1/(1/x) = x $

On a graph, the curve of an inverse function $ f^{(-1)} $ is the symmetrical curve of the curve $ f $ with respect to the diagonal axis $ y = x $

The inverse function of a constant function $ f(x) = a $ is the linear function of equation $ x = a $

For a function to have a reciprocal function on an interval, it must be bijective, continuous and strictly monotonic on this interval.

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Cite as source (bibliography):

*Reciprocal Function* on dCode.fr [online website], retrieved on 2024-07-18,

- Reciprocal/Inverse Function Calculator
- What is a reciprocal function? (Definition)
- How to calculate an inverse function?
- Is the inverse function of a function unique?
- What function is its own reciprocal function?
- How to graphically plot a reciprocal function?
- What is the reciprocal of a constant function?
- What are the conditions for the existence of an inverse function?

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