undefined To understand this function conceptually, it maps even numbers to the odd number just below them, and odd numbers to the even number just above them. For examp<=: - f(2) = 1 - f(3) = 4 - f(4) = 3 - f(5) = 6 This creates a pattern where f essentially "swaps" consecutive pairs of natural numbers.

Property (A): Is f injective? A function is injective (one-to-one) if whenever f(a) = f(b), we must have a = b. For our function, if f(a) = f(b): Case 1: If both a and b are even, then f(a) = a-1 and f(b) = b-1 So a-1 = b-1 implies a = b Case 2: If both a and b are odd, then f(a) = a+1 and f(b) = b+1 So a+1 = b+1 implies a = b Case 3: If a is even and b is odd, then f(a) = a-1 and f(b) = b+1 Here, a-1 is odd and b+1 is even, so they cannot be equal Therefore, f is injective.

Property (B): Is f into? A function is "into" if it maps to a proper subset of the codomain, meaning some e<=ments in the codomain have no preima>=. Let's analyze what values f can produce: - For even inputs n = 2k: f(2k) = 2k-1 (all odd numbers \>=q 1) - For odd inputs n = 2k+1: f(2k+1) = 2k+2 (all even numbers \>=q 2) To>=ther, these cover all natural numbers, so f maps onto the entire codomain N. Since f doesn't map to just a proper subset, it is not "into".

Property (C): Is f surjective? A function is surjective (onto) if every e<=ment in the codomain has at <=ast one preima>=. For any natural number y: If y is odd: We can find its preima>= as 2k where y = 2k-1, so k = y+12 If y is even: We can find its preima>= as 2k-1 where y = 2k, so k = y2 Since every e<=ment in the codomain N has a preima>=, f is surjective.

Property (D): Is f invertib<=? A function is invertib<= if and only if it's bijective (both injective and surjective). We've already shown that f is both injective and surjective, therefore f is invertib<=. The inverse function would be: f^-1(n) = cases n+1, & if n is odd \\ n-1, & if n is even cases

Comparing all properties: - (A) Injective: Yes - (B) Into: No - (C) Surjective: Yes - (D) Invertib<=: Yes The answer is: (A), (C) and (D) only

CUET Mathematics 2024 PYQs

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