Inverse function
lastmod 2023-01-09created 2023-01-096 edits
A function \(f\) and its inverse function \(f^{-1}\)
In mathematics, an inverse function (Hanja: 逆函數; North Korean standard language: geokkul hamsu; English: inverse function) is a function obtained by swapping the domain and the range (function values). That is, in the correspondence rule of the inverse function, the original output values correspond to the original input values.
How to find an inverse function
Finding an inverse function is simpler than you might think. You just swap the domain and the range.
- Check whether the function
y = f(x)is a one-to-one correspondence. - Solve \(y = f(x)\) for x -> \(x = f^{-1}(y)\)
- Swap x and y -> \(y = f^{-1}(x)\)
- Swap the domain and range of the function f.
Examples of one-to-one correspondence
To find an inverse function, first check whether it is a one-to-one correspondence, solve for x, then swap x and y. After that you need to swap the domain and the range, but since the functions in these problems have the set of all real numbers as both domain and codomain, you don't need to worry much about it here.
\(y = x + 1\) is indeed a one-to-one correspondence. Let's find its inverse:
- \(y = x + 1\)
- \(x = y - 1\)
- \(y = x - 1\)
\(y = x^{2}+1\) is not a one-to-one correspondence, so its inverse cannot be found.
- When
x = 1,y = 2 - When
x = -1,y = 2