Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if there exists a function g from Y to X such that $${\displaystyle g(f(x))=x}$$ for all $${\displaystyle x\in X}$$ and See more

Squaring and square root functions
The function f: R → [0,∞) given by f(x) = x is not injective because $${\displaystyle (-x)^{2}=x^{2}}$$ for all $${\displaystyle x\in \mathbb {R} }$$. Therefore, f is not invertible.
If the domain of the … See more

• Let f be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit, $${\displaystyle F=f(C)={\tfrac {9}{5}}C+32;}$$ then … See more

• Briggs, William; Cochran, Lyle (2011). Calculus / Early Transcendentals Single Variable. Addison-Wesley. ISBN 978-0-321-66414-3 See more

Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.
Uniqueness See more

Partial inverses
Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by See more

• Lagrange inversion theorem, gives the Taylor series expansion of the inverse function of an analytic function
• Integral of inverse functions
• Inverse Fourier transform See more