What is an example of an injective only function?

Injective functions are a specific type of mathematical mappings where each element in the domain corresponds uniquely to an element in the codomain. A prime example of an injective function is the identity function, denoted as X → X, which maps each element to itself, thus ensuring a one-to-one correspondence.

Another illustrative example of an injective function is f: R→ R, where f(x) = 2x. Here, for any real number x, doubling it results in a unique output, ensuring that no two distinct inputs map to the same output. This characteristic of uniqueness defines the injectivity of the function.

Similarly, the function f: R→ R, where f(x) = 2x + 1, is also injective. By adding a constant 1 to the doubled input, we maintain the one-to-one relationship between the domain and the codomain, ensuring that each input maps to a distinct output.

However, not all functions exhibit injectivity. For instance, the function f: R→ R, where f(x) = x², is not injective. This is because, for example, both -1 and 1 map to the same output, namely 1. Thus, the function fails to satisfy the injectivity criterion, which requires each input to have a unique output.

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