Let f1 : R → R, f2 : [0, \infty) → R, f3 : R → R and f4 : R → [0, \infty) be defined by f1(x) = \left{\right. \left|\right. x \left|\right. , if \textrm{ } x < 0 \\ e^{x} , if \textrm{ } x \geq 0 f2(x) = x2, f3(x) = \left{\right. sinx , if \textrm{ } x < 0 \\ x , if \textrm{ } x \geq 0 and f_{4} \left(\right. x \left.\right) = \left{\right. f_{2} \left(\right. \textrm{ } f_{1} \left(\right. x \left.\right) \textrm{ } \left.\right) , if \textrm{ } x < 0 \\ f_{2} \left(\right. \textrm{ } f_{1} \left(\right. x \left.\right) \textrm{ } \left.\right) - 1 , if \textrm{ } x \geq 0 List I List II (P) f4 is (1) onto but not one‑one (Q) f3 is (2) neither continuous nor one‑one (R) f2of1 is (3) differentiable but not one‑one (S) f2 is (4) continuous and one‑one

Engineering

Mathematics

Important Points on Derivability

Question

Let f₁ : R → R, f₂ : [0, $\infty$ ) → R, f₃ : R → R and f₄ : R → [0, $\infty$ ) be defined by

f₁(x) = ${\begin{cases} | x |, if x < 0 \\ e^{x}, if x \geq 0 \end{cases}$

f₂(x) = x²,

f₃(x) = ${\begin{cases} sinx, if x < 0 \\ x, if x \geq 0 \end{cases}$

and

$f_{4} (x) = {\begin{cases} f_{2} (f_{1} (x)), if x < 0 \\ f_{2} (f_{1} (x)) - 1, if x \geq 0 \end{cases}$

List I	List II
(P) f₄ is	(1) onto but not one‑one
(Q) f₃ is	(2) neither continuous nor one‑one
(R) f₂of₁ is	(3) differentiable but not one‑one
(S) f₂ is	(4) continuous and one‑one

P → 1

Q → 3

R → 4

S → 2

P → 3

Q → 1

R → 4

S → 2

P → 1

Q → 3

R → 2

S → 4

P → 3

Q → 1

R → 2

S → 4

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f₁(x) = ${\begin{cases} - x, if x < 0 \\ e^{x}, if x \geq 0 \end{cases}$

f₂(x) = x²,

f₃(x) = ${\begin{cases} sinx, if x < 0 \\ x, if x \geq 0 \end{cases}$

and

f₄(x) = ${\begin{cases} f_{2} (f_{1} (x)), if x < 0 \\ f_{2} (f_{1} (x)) - 1, if x \geq 0 \end{cases}$

Now verify.