An inverse function reverses the operation of a given function, mapping outputs back to their original inputs, provided the function is bijective (both injective and surjective). The notation for an inverse function is typically f⁻¹(x), and it satisfies the condition that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain of the inverse.

Inverse Function

inverse function

operation of a given function

mapping outputs back

original inputs

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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