What is not an injective?

In mathematics, the concept of injectivity, also known as being one-to-one, plays a crucial role in understanding functions. A function fails to be injective when it maps at least two distinct inputs to the same output. This phenomenon contradicts the fundamental principle of a one-to-one correspondence, where each input is uniquely associated with a single output.

To illustrate this point, consider the function f(x) = x^2. This function, commonly known as the squaring function, operates by squaring the value of its input, x. Despite its simplicity, it exhibits a notable property that renders it non-injective.

Specifically, for the function f(x) = x^2, there exist two distinct inputs, namely 1 and -1, that yield the same output. When we substitute 1 into the function, we get f(1) = 1^2 = 1. Similarly, substituting -1 gives us f(-1) = (-1)^2 = 1. Both inputs produce the output 1, demonstrating the function's failure to be injective.

The essence of this non-injectivity lies in the fact that the squaring function treats positive and negative numbers equally when it comes to their magnitudes. Consequently, any two numbers with opposite signs but the same absolute value will map to the same square, violating the one-to-one nature of an injective function.

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