Let $f_1 : R \to R$, $f_2 : [0, \infty) \to R$, $f_3 : R \to R$ and $f_4 : R \to [0, \infty)$ be defined by $f_1(x) = \begin{cases} |x| & \text{if } x < 0, \\ e^{x} & \text{if } x \ge 0; \end{cases}$ $f_2(x) = x^{2}$; $f_3(x) = \begin{cases} \sin x & \text{if } x < 0, \\ x & \text{if } x \ge 0; \end{cases}$ and $f_4(x) = \begin{cases} f_2(f_1(x)) & \text{if } x < 0, \\ f_2(f_1(x)) - 1 & \text{if } x \ge 0. \end{cases}$ | List-I | List-II | | --- | --- | | (P) $f_4$ is | (1) Onto but not one-one | | (Q) $f_3$ is | (2) Neither continuous nor one-one | | (R) $f_2 \circ f_1$ is | (3) Differentiable but not one-one | | (S) $f_2$ is | (4) Continuous and one-one |

Let f₁ : R → R, f₂ : [0, ∞) → R, f₃ : R → R and f₄ : R → [0, ∞) be defined by 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. | List-I | List-II | | --- | --- | | (P) f₄ is | (1) Onto but not one-one | | (Q) f₃ is | (2) Neither continuous nor one-one $f_1(x) = \begin{cases} |x| & x<0 \\ e^x & x\ge 0 \end{cases} \quad f_2(x) = x^2$ $f_3(x) = \begin{cases} \sin x & x<0 \\ x & x\ge 0 \end{cases} \quad f_4(x) = \begin{cases} f_2(f_1(x)) & x<0 \\ f_2(f_1(x))-1 & x\ge 0 \end{cases}$ $\text{For each: continuity, differentiability, one-one / onto}$ $\text{check the break point } x=0$ $(f_2\circ f_1)(x) = \begin{cases} (-x)^2 = x^2 & x<0 \\ (e^x)^2 = e^{2x} & x\ge 0 \end{cases}$ $\lim_{x\to 0^-}x^2 = 0 \neq 1 = \lim_{x\to 0^+}e^{2x} \Rightarrow \text{discontinuous}$ $(R) \to (2)\ \text{Neither continuous nor one-one}$ $f_3(0)=0,\quad \lim_{x\to0}\sin x = 0 = \lim_{x\to0}x \Rightarrow \text{continuous}$ $\text{left derivative } \cos 0 = 1 = \text{right derivative}$ $f_3 \text{ not one-one (sin repeats on } x<0)$ $(Q) \to (3)\ \text{Differentiable but not one-one}$ $f_2(x)=x^2 \text{ on } [0,\infty):\ \text{continuous, strictly increasing}$ $(S) \to (4)\ \text{Continuous and one-one}$ $f_4(x) = \begin{cases} x^2 & x<0 \\ e^{2x}-1 & x\ge 0 \end{cases}$ $\lim_{x\to0^-}x^2 = 0 = \lim_{x\to0^+}(e^{2x}-1) \Rightarrow \text{continuous}$ $\text{range} = [0,\infty) = \text{codomain} \Rightarrow \text{onto; not one-one}$ $(P) \to (1)\ \text{Onto but not one-one}$ $P\to1;\ Q\to3;\ R\to2;\ S\to4$ $\boxed{\text{Answer: option (4)}}$