Question

Let \(f: \R → \R\) be defined by \(f(x) = x^2\), for every \(x∈\R\). Then \(f\) is:

• A. one-one and onto
• B. one-one and not onto
• C. neither one-one nor onto
• D. onto and not one-one

Answer 1

The Correct Option is C

To determine the characteristics of the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\), consider the following properties:

One-One (Injective): A function \(f\) is one-one if \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\) for all \(x_1, x_2 \in \mathbb{R}\). For \(f(x) = x^2\), assume \(f(x_1) = f(x_2)\) which leads to \(x_1^2 = x_2^2\). This implies either \(x_1 = x_2\) or \(x_1 = -x_2\). Since it's possible for \(x_1 \neq x_2\) (e.g., \(x_1 = 1, x_2 = -1\)), the function is not one-one.

Onto (Surjective): A function \(f\) is onto if for every \(y \in \mathbb{R}\), there exists \(x \in \mathbb{R}\) such that \(f(x) = y\). Consider \(y = -1\). There is no real number \(x\) such that \(x^2 = -1\) because the square of any real number is non-negative. Hence, not all elements in \(\mathbb{R}\) have a preimage, so \(f\) is not onto.

Conclusion: The function \(f(x) = x^2\) is neither one-one nor onto.

Options	Conclusion
one-one and onto	No
one-one and not onto	No
neither one-one nor onto	Yes
onto and not one-one	No