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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.

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Reciprocal Function -

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# Reciprocal Function

## Reciprocal/Inverse Function Calculator

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

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

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 $x = a$

## Source code

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