To address this, let’s analyze the function [tex]\( f: \mathbb{R} \to \mathbb{R} \)[/tex] defined by [tex]\( f(x) = x^2 \)[/tex].
### Checking if [tex]\( f(x) = x^2 \)[/tex] is One-to-One (Injective)
A function is one-to-one (injective) if different inputs produce different outputs. In other words, [tex]\( f(x_1) = f(x_2) \)[/tex] implies [tex]\( x_1 = x_2 \)[/tex].
Consider the function [tex]\( f(x) = x^2 \)[/tex]:
- Let's assume [tex]\( x_1 \neq x_2 \)[/tex]. However, if [tex]\( f(x_1) = f(x_2) \)[/tex], these would give us [tex]\( x_1^2 = x_2^2 \)[/tex].
- This could be true in two cases: [tex]\( x_1 = x_2 \)[/tex] or [tex]\( x_1 = -x_2 \)[/tex].
Since [tex]\( f(x) = x^2 \)[/tex] gives the same result for both a positive and its negative counterpart, such as [tex]\( f(3) = 9 \)[/tex] and [tex]\( f(-3) = 9 \)[/tex], it means that [tex]\( f(x_1) = f(x_2) \)[/tex] does not necessarily imply [tex]\( x_1 = x_2 \)[/tex]. Thus, the function [tex]\( f(x) = x^2 \)[/tex] is not one-to-one.
### Checking if [tex]\( f(x) = x^2 \)[/tex] is Onto (Surjective)
A function is onto (surjective) if every element of the function’s codomain (here, [tex]\(\mathbb{R}\)[/tex]) is the image of at least one element from its domain (also [tex]\(\mathbb{R}\)[/tex]).
Consider the range of [tex]\( f(x) = x^2 \)[/tex]:
- [tex]\( f(x) = x^2 \)[/tex] outputs only non-negative values since squaring any real number [tex]\( x \)[/tex] always results in a non-negative value.
- Thus, [tex]\( f(x) \geq 0 \)[/tex] for all [tex]\( x \in \mathbb{R} \)[/tex].
Hence, not every real number [tex]\( y \in \mathbb{R} \)[/tex] has a preimage in [tex]\( \mathbb{R} \)[/tex] because real negative numbers do not have corresponding real inputs. For instance, there is no real [tex]\( x \)[/tex] such that [tex]\( f(x) = x^2 = -1 \)[/tex].
Thus, the function [tex]\( f(x) = x^2 \)[/tex] is not onto.
### Conclusion
Having established that [tex]\( f(x) = x^2 \)[/tex] is neither one-to-one nor onto, we conclude that the correct answer to the given problem is:
[tex]\[ \boxed{\text{(D) neither one-one nor onto}} \][/tex]
