Proving x squared is not injective

Prove that the function $f(x) = x^2$ is not injective.

The concept of injectivity, also known as one-to-one, is important in mathematics, especially in the study of functions and mappings. A function is said to be injective if every element of the function's codomain is mapped to by at most one element of its domain. In other words, no two different elements in the domain of the function map to the same element in the codomain. Understanding injectivity involves analyzing the behavior of a function in terms of its ability to uniquely associate an element of its domain with an element of its codomain.

To determine if the function $f(x) = x^2$ is injective, one needs to apply this concept. For a function to fail being injective, there need to be at least two distinct inputs that produce the same output. In the case of $f(x) = x^2$, this can happen when negative and positive versions of a number yield the same squared result, such as when both -2 and 2 result in 4. This characteristic can be generalized for any real number, since $f(a) = f(-a)$.

When studying functions and mappings in mathematics, it's crucial to understand which functions have unique outputs for every input and which do not. Conclusive proof that certain types of functions are not injective helps in fields requiring bijective (both injective and surjective) functions, such as solving equations or moving between coordinate systems, and in computational processes that rely on function properties for performance optimization. Understanding these concepts allows mathematicians and students to approach more complex problems in function analysis with a strong foundational knowledge of these basic, yet critical properties.