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
$f_{1} (x) = {\begin{matrix} | x | if x < 0, e^{x} if x \geq 0; \end{matrix}$

$f_{2} (x) = x^{2};$

$f_{3} (x) = {\begin{matrix} sin x if x < 0, x if x \geq 0; \end{matrix}$

and

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

$\begin{matrix} L i s t I & L i s t I I P. f_{4} is & 1. onto but not one-one Q. f_{3} is & 2. neither continuous nor one-one R. f_{2} \circ f_{1} is & 3. differentiable but not one-one S. f_{2} is & 4. continuous and not one-one \end{matrix}$

which of the following option is correct ?

(P) \to (3), (Q) \to (1) (R) \to (4), (S) \to (2)

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(P) \to (1), (Q) \to (3) (R) \to (4), (S) \to (2)

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(P) \to (3), (Q) \to (1) (R) \to (2), (S) \to (4)

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(P) \to (1), (Q) \to (3) (R) \to (2), (S) \to (4)

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Solution

The correct option is D $(P) \to (1), (Q) \to (3) (R) \to (2), (S) \to (4)$
$f_{4} (x) = {\begin{matrix} f_{2} (f_{1} (x)) if x < 0, f_{2} (f_{1} (x)) - 1 if x \geq 0. \end{matrix} \Rightarrow f_{4} (x) = {\begin{matrix} x^{2} if x < 0, e^{2 x} - 1 if x \geq 0. \end{matrix}$

As $f_{4} : R \to [0, \infty)$
onto but not one-one

$f_{3} (x) = {\begin{matrix} sin x if x < 0, x if x \geq 0; \end{matrix}$

We know that,
$f_{3}^{'} (x) = {\begin{matrix} cos x if x < 0, 1 if x \geq 0; \end{matrix} \Rightarrow f_{3}^{'} (0) = {\begin{matrix} 1 if x < 0, 1 if x \geq 0; \end{matrix}$
differentiable but not one-one

$f_{2} \circ f_{1} (x) = {\begin{matrix} x if x < 0, e^{2 x} if x \geq 0. \end{matrix}$
neither continuous nor one-one

$f_{2} (x) = x^{2}$
The domian of function is $[0, \infty)$
Therefore, continuous and one-one

Hence,
$(P) \to (1), (Q) \to (3) (R) \to (2), (S) \to (4)$

Suggest Corrections

Similar questions

Match the conditions/expressions in Column I with statement in Column II.

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 $f_{1} (x) = {\begin{matrix} | x |, i f x < 0 e^{x}, i f x \geq 0 \end{matrix} f_{2} (x) = x^{2}; f_{3} (x) = {\begin{matrix} s i n x, i f x < 0 x, i f x \geq 0 \end{matrix}$ and $f_{4} (x) = {\begin{matrix} f_{2} [f_{1} (x)], i f x < 0 f_{2} [f_{1} (x)] - 1, i f x \geq 0 \end{matrix}$
$\begin{matrix} Column I & Column II a . f_{4} i s & p . onto but not one-one b . f_{3} i s & q . neither continuous nor one-one c . f_{2} o f f_{1} i s & r . differentiable but not one-one d . f_{2} i s & s . continuous and one-one \end{matrix}$

Q. Let

f_{1} : R \to R, f_{2} : (- \frac{π}{2}, \frac{π}{2}) \to R, f_{3} : (- 1, e^{\frac{π}{2}} - 2) \to R

and

f_{4} : R \to R

be functions defined by
(i)

f_{1} (x) = sin (\sqrt{1 - e^{- x^{2}}})

(ii)

f_{2} (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{| sin x |}{{tan}^{- 1} (x)} & i f & x \neq 0 1 & i f & x = 0 \end{matrix}

where the inverse trigonometric function

{tan}^{- 1} (x)

assumes values in

(- \frac{π}{2}, \frac{π}{2})

.
(iii)

f_{3} (x) = [sin ({log}_{e} (x + 2)]

, where for

t ϵ R, [t]

denotes the greatest integer less than or equal to

t

,
(iv)

f_{4} (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} x^{2} sin (\frac{1}{x}) & i f x \neq 0 0 & i f x = 0 \end{matrix}

List - I	List - II
P. The function $f_{1}$ is	$1.$ NOT continuous at $x = 0$
Q. The function $f_{2}$ is	$2.$ continuous at $x = 0$ and NOT differentiable at $x = 0$
R. The function $f_{3}$ is	$3.$ differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
S. The function $f_{4}$ is	$4.$ diffferentiable at $x = 0$ and its derivative is continuous at $x = 0$

The correct option is