Inverse function

An inverse function reverses a function’s input-output mapping, sending each output back to its unique original input. It exists as a function only when the original mapping is one-to-one on its domain and has the required range.

One-to-one function: A function that maps distinct inputs to distinct outputs. Only one-to-one functions have unique original inputs for every output.
Function: A rule assigning exactly one output to each input in its domain. An inverse function reverses this input-output assignment.
Inverse trigonometric functions: Functions that return angles from trigonometric values, defined on restricted domains to ensure uniqueness. Restricting sine, cosine, or tangent makes their inverses functions.
Reciprocal function: The function that maps each nonzero number x to 1/x. A reciprocal is often called a multiplicative inverse, but it does not generally undo a function.
Horizontal line test: A graphical test for whether a function is one-to-one: no horizontal line intersects its graph more than once. It reveals whether the original function can be reversed without ambiguity.
Injective function: A function in which distinct inputs always have distinct outputs. Injectivity is the precise condition that prevents two inputs sharing an inverse output.
Logarithm: The inverse of an exponential function, giving the exponent required to produce a specified positive value. Logarithms undo exponentiation when solving equations.
Inverse image: The set of inputs that a function maps into a specified subset of its codomain. An inverse image exists for any function, even when an inverse function does not.
Function composition: An operation that applies one function to the output of another. Composing a function with its inverse returns the original input.
Surjective function: A function whose range equals its specified codomain. Surjectivity ensures every element of the codomain has an inverse image.

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