Search for a tool
Reciprocal Function

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.

Results

Reciprocal Function -

Tag(s) : Functions

Share
dCode and more

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!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Reciprocal Function' tool for free! Thank you!

# Reciprocal Function

## Reciprocal/Inverse Function Calculator

See also: Equation Solver

## Answers to Questions (FAQ)

### What is a reciprocal function? (Definition)

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.

### How to calculate an inverse function?

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)$

### Is the inverse function of a function unique?

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

### What function is its own reciprocal function?

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

Example: $f(1/x) = 1/(1/x) = x$

### How to graphically plot a reciprocal function?

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 $y = x$

### What is the reciprocal of a constant function?

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

### What are the conditions for the existence of an inverse function?

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

## Source code

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.

## Cite dCode

The copy-paste of the page "Reciprocal Function" or any of its results, is allowed (even for commercial purposes) as long as you credit 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 2024-07-18, https://www.dcode.fr/reciprocal-function

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

## Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Reciprocal Function' tool for free! Thank you!

https://www.dcode.fr/reciprocal-function
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback