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Consider the functions f₁: R → R, f₂: [0, ∞) → R, f₃: R → R, and f₄: R → [0, ∞) defined as follows: f₁(x) = |x| if x < 0, eˣ if x ≥ 0; f₂(x) = x²; f₃(x) = sin(x) if x < 0, x if x ≥ 0; and f₄(x) = f₂(f₁(x)) if x < 0, f₂(f₁(x)) − 1 if x ≥ 0. Match the descriptions of these functions in List I with the properties in List II.

1. P corresponds to 3, Q to 1, R to 4, and S to 2
2. P corresponds to 1, Q to 3, R to 4, and S to 2
3. P corresponds to 3, Q to 1, R to 2, and S to 4
4. P corresponds to 1, Q to 3, R to 2, and S to 4

Correct answer: P corresponds to 1, Q to 3, R to 2, and S to 4

Solution

Matching the functions with their properties involves analyzing their definitions. For P, f₁(x) is continuous but not differentiable at x = 0. For Q, f₂(x) is differentiable everywhere. For R, f₃(x) is continuous but not differentiable at x = 0. For S, f₄(x) is continuous and differentiable everywhere except at x = 0. This matches option D.

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