Why f(f-1(x))=x?

The equality f(f⁻¹(x)) = x is a direct consequence of the definition of an inverse function, but it holds only under specific conditions. For a function f to have an inverse f⁻¹, it must be bijective—that is, both injective (one-to-one) and surjective (onto) over the domains in question. Injectivity ensures that no two distinct inputs map to the same output, which is necessary for the inverse mapping to be a function itself. Surjectivity ensures that every element in the codomain is an output of f, guaranteeing that the inverse is defined for every element in that codomain. When these conditions are met, the inverse function f⁻¹ is formally defined as the function that, given an output y of f, returns the unique input x such that f(x) = y. The composition f(f⁻¹(x)) then asks for the output of f when given the input that specifically produces x. By definition, applying f to that input must yield x, confirming the identity.

The mechanism is best understood by tracing an element through the mappings. Starting with a value x in the codomain of f, the inverse function f⁻¹ identifies the unique element a in the domain such that f(a) = x. Therefore, f⁻¹(x) = a. The subsequent composition applies f to this result: f(f⁻¹(x)) = f(a). Since a was chosen precisely because f(a) = x, the output is x. This process effectively reverses and then re-applies the original mapping, returning to the starting point. It is a precise algebraic manifestation of the symmetric relationship between a function and its inverse; each undoes the action of the other. This property is foundational and is often used as part of the formal definition of an inverse function, alongside the complementary identity f⁻¹(f(x)) = x, which operates in the reverse direction on the domain.

A critical nuance is that the domain of x in the expression f(f⁻¹(x)) = x is exactly the range of the original function f, which is also the domain of f⁻¹. If f is not surjective onto the intended codomain, the inverse may only be defined on a restricted set, and the equality holds only for x in that set. For example, if f: ℝ → ℝ is defined by f(x) = x², it is not injective over all reals and thus does not have a general inverse. If we restrict its domain to non-negative numbers and define f⁻¹ as the principal square root function, then for any x ≥ 0, f(f⁻¹(x)) = (√x)² = x. For negative x, f⁻¹(x) is not defined within the real numbers under this restriction, so the equality is not applicable. This highlights that the identity is not a universal algebraic trick but a conditional property of a properly defined inverse pair.

The implications of this identity are pervasive in mathematics and its applications. In algebra, it underpins solving equations, as applying f⁻¹ to both sides of f(a) = b yields a solution. In calculus, it is essential for derivative rules involving inverse functions, such as (f⁻¹)′(x) = 1 / f′(f⁻¹(x)), which relies on the functional relationship. In computer science, invertible functions are crucial for cryptography and data encoding, where the ability to reliably recover an original input from an output is required. The identity f(f⁻¹(x)) = x serves as a fundamental test for whether a proposed inverse is correct and ensures the consistency of mathematical operations involving functional inversion. Its validity is the bedrock upon which the utility of inverse functions is built.