Question

Consider the function \( f : \mathbb{R} \rightarrow \mathbb{R} \), defined as \( f(x) = x^3 \). Assertion (A): \( f(x) \) is a one-one function. Reason (R): \( f(x) \) is a one-one function, if co-domain = range.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

A function is one-one if it passes the horizontal line test. The condition for one-one does not depend on the co-domain equaling the range, but on the uniqueness of outputs for distinct inputs.

Answer 1

The Correct Option is C

To determine the correctness of the Assertion (A) and Reason (R), we analyze the given function \( f(x) = x^3 \). - Assertion (A): \( f(x) \) is a one-one function. A function is one-one (injective) if different inputs produce different outputs, i.e., if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \). For \( f(x) = x^3 \), if \( f(x_1) = f(x_2) \), then \( x_1^3 = x_2^3 \). Since the cubic function is strictly increasing (its derivative \( f'(x) = 3x^2 \geq 0 \) and \( f'(x) = 0 \) only at \( x = 0 \), where it changes from decreasing to increasing but remains one-one), \( x_1 = x_2 \). Thus, \( f(x) = x^3 \) is one-one. Assertion (A) is true. - Reason (R): \( f(x) \) is a one-one function, if co-domain = range. A function is one-one if it is injective, which depends on the mapping of inputs to outputs, not necessarily on the co-domain equaling the range. For \( f(x) = x^3 \), the range is all real numbers \( \mathbb{R} \) (since \( x^3 \) covers all \( \mathbb{R} \) as \( x \) varies over \( \mathbb{R} \)), and the co-domain is \( \mathbb{R} \). However, the condition "co-domain = range" is not a requirement for a function to be one-one; it is a condition for the function to be onto (surjective). A function can be one-one even if the co-domain is larger than the range. Thus, Reason (R) is false. - Conclusion: Assertion (A) is true, but Reason (R) is false. The correct option is (C). \[ \boxed{\text{(C)}} \]