Question

Let $\mathbb{N}, \mathbb{Z}$ and $\mathbb{R}$ be the set of natural numbers, integers and real numbers respectively, $[\cdot]$ denotes the greatest integer function. Match List-I with List-II:}

• A. (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
• B. (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
• C. (A) - (III), (B) - (I), (C) - (II), (D) - (IV)
• D. (A) - (II), (B) - (IV), (C) - (I), (D) - (III)

Hint:

Linear functions with non-zero slope are always one-one and onto over $\mathbb{R}$.

Answer 1

The Correct Option is A

Concept: We classify functions based on injectivity (one-one) and surjectivity (onto).

Step 1: {Analyze (A) $f(x)=x^2$ on $\mathbb{N} \to \mathbb{N}$.}
Since domain is natural numbers: \[ x_1^2=x_2^2 \Rightarrow x_1=x_2 \] So function is one-one. But not onto because not all natural numbers are perfect squares. \[ \Rightarrow (II) \]

Step 2: {Analyze (B) $f(x)=x^2$ on $\mathbb{R} \to \mathbb{R}$.}
\[ f(x)=f(-x) \Rightarrow not\ one-one \] Also negative numbers are not covered in range. So neither one-one nor onto: \[ \Rightarrow (IV) \]

Step 3: {Analyze (C) $f(x)=2x+3$.}
Linear function with non-zero slope: \[ f(x_1)=f(x_2)\Rightarrow x_1=x_2 \] So one-one. Also every real number can be obtained: \[ y=2x+3 \Rightarrow x=\frac{y-3}{2} \] So onto. \[ \Rightarrow (I) \]

Step 4: {Analyze (D) $f(x)=[x]$.}
Floor function maps many reals to same integer → not one-one. But every integer is achieved → onto. \[ \Rightarrow (III) \]