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

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

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.

dCode retains ownership of the "Reciprocal Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Reciprocal Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Reciprocal Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Reciprocal Function" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Reciprocal Function" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Reciprocal Function* on dCode.fr [online website], retrieved on 2023-09-27,

- 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?

inverse,function,reciprocal,equation,bijective

https://www.dcode.fr/reciprocal-function

© 2023 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback