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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Reciprocal Function
Reciprocal/Inverse Function Calculator
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.
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- 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?
