Analyzing the Function f(x) x2: Injectivity and Surjectivity

When examining the function f : R → R defined by f(x) x2, it is important to understand whether it is injective or surjective. This article explores these properties in detail, providing a comprehensive analysis using mathematical reasoning and examples.

Injectivity: A Function's One-to-One Nature

A function f is considered injective or one-to-one if distinct inputs yield distinct outputs. In mathematical terms, f(x1) f(x2) implies x1 x2 for all x1, x2 ∈ R. The function f(x) x2 is not injective because we can find two different inputs that produce the same output.

For instance, let's consider the inputs x1 1 and x2 -1: f(1) 12 1 f(-1) (-1)2 1 Here, f(1) f(-1) but 1 ≠ -1. The presence of such distinct inputs leading to the same output indicates that the function is not injective.

Surjectivity: Covering the Entire Codomain

A function f is surjective or onto if every element in the codomain can be achieved as an output for some input from the domain. In mathematical terms, for every y ∈ R, there exists at least one x ∈ R such that f(x) y. For f(x) x2, this property does not hold, and the function is not surjective.

Consider the output of f(x) x2, which is always non-negative (i.e., f(x) ≥ 0 for all x). This implies that there are no values of x that can produce y . For example, if we try to find an x such that f(x) -1, we get: x2 -1 There is no real number x that satisfies this equation, meaning that ?1 is not in the range of the function f.

Conclusion and Further Insights

In summary, the function f(x) x2 fails both injectivity and surjectivity tests. The function is not injective because distinct inputs can produce the same output, and it is not surjective because it cannot produce all values in the codomain.

Additional Examples and Considerations

To further illustrate, consider the following examples:

If x 2, then f(2) 4; if x -2, then f(-2) 4. Therefore, f is not injective. The product of any two positive numbers is positive, and the product of two negative numbers is also positive. Consequently, -1 is not in the range of f. Hence, f is not surjective. It is worth noting that if we restrict the domain and range to [0, ∞), the function f(x) x2 becomes both injective and surjective.

This analysis underscores the importance of carefully examining the properties of functions to fully understand their behavior and limitations.