Let N, Z and R be the set of natural numbers, set of integers and set of real numbers respectively, [.] denotes the greatest integer function then
Match List-I with List-II
| List-I | List-II |
| (A) \( f:N\to N,f(x)=x^2 \) | (I) One-one and onto |
| (B) \( f:R\to R,f(x)=x^2 \) | (II) One-one but not onto |
| (C) \( f:R\to R,f(x)=2x+3 \) | (III) Not one-one but onto |
| (D) \( f:R\to Z,f(x)=[x] \) | (IV) Neither one-one nor onto |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (II), (B) - (IV), (C) - (I), (D) - (III)
(A) \( f:N\to N,\ f(x)=x^2 \)
This matches (II) One-one but not onto. Different natural numbers have different squares, but 2 is not a square.
(B) \( f:R\to R,\ f(x)=x^2 \)
This matches (IV) Neither one-one nor onto. f(−1) = f(1), and negative numbers are not squares.
(C) \( f:R\to R,\ f(x)=2x+3 \)
This matches (I) One-one and onto. Each real y comes from exactly one x, namely \( x=\frac{y-3}{2} \).
(D) \( f:R\to Z,\ f(x)=[x] \)
This matches (III) Not one-one but onto. [1.2] = [1.5] = 1, but every integer n equals [n].
So the matching is (A) - (II), (B) - (IV), (C) - (I), (D) - (III).