Inverse functions reverse the effect of the original function, mapping outputs back to their corresponding inputs, and exist only when the function is bijective (both injective and surjective). The graph of an inverse function is a reflection of the original function's graph across the line y = x.

Inverse Functions

Inverse functions

original function

A bijective function is a mathematical function that is both injective (one-to-one) and surjective (onto), meaning each element of the function's domain maps to a unique element of its codomain, and every element of the codomain is mapped by some element of the domain. This property ensures that a bijective function has an inverse function, which uniquely reverses the mapping process.

Bijective Function

An injective function, also known as a one-to-one function, ensures that distinct inputs map to distinct outputs, meaning no two different elements in the domain are mapped to the same element in the codomain. This property is crucial for establishing a function's invertibility on its image, as it guarantees a unique inverse function can be defined for the range of the injective function.

Injective Function

A surjective function, or onto function, is a type of mapping between two sets where every element in the codomain has at least one pre-image in the domain. This ensures that the function covers the entire codomain, making it an essential concept in understanding the structure and behavior of mathematical functions.

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