NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 2

Question

Check injectivity and surjectivity of (i) f:ℕ→ℕ, f(x)=x²; (ii) f:ℤ→ℤ, f(x)=x²; (iii) f:ℝ→ℝ, f(x)=x²; (iv) f:ℕ→ℕ, f(x)=x³; (v) f:ℤ→ℤ, f(x)=x³.

Accepted Answer

Final Answer: (i),(iv),(v) injective but not onto; (ii),(iii) neither. Step 1: (i) For positive a,b, a²=b² implies a=b. But 2∈ℕ is not a square of a natural number, so not onto. Step 2: (ii) f(1)=f(−1)=1 with distinct integer inputs, so not injective. Negative integers are not squares, so not onto. Step 3: (iii) The same input pair disproves injectivity, and −1 has no real square root, disproving onto ℝ. Step 4: (iv) Cubing is strictly increasing on ℕ, so injective. The natural number 2 is not a natural cube, so not onto. Step 5: (v) Cubing is strictly increasing on ℤ, so injective. The integer 2 is not an integer cube, so not onto.

NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 2

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