Question

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
$f_1(x)=\{\begin{array}{llc}|x|& \text{ if }& x<0\\ e^x& \text{ if }& x\ge 0\end{array};f_2(x)=x^2;f_3(x)=\{\begin{array}{llc}\sin x& \text{ if }& x<0\\ x& \text{ if }& x\ge 0\end{array}$ and $f_4(x)=\{\begin{array}{llc}{f}_2\left(f_1(x)\right)& \text{ if }& x<0\\ f_2\left(f_1(x)\right)-1& \text{ if }& x\ge 0\end{array}$

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	$f_{2}o{f}_1$ is	3	differentiable but not one-one
S	$f_2$ is	4	continuous and one-one

• A. P - (3) Q - (1) R - (4) S - (2)
• B. P - (1) Q - (3) R - (4) S - (2)
• C. P - (3) Q - (1) R - (2) S - (4)
• D. P - (1) Q - (3) R - (2) S - (4)

Answer 1

Answer: (D)

$f_2\left(f_1\right)=\{\begin{array}{ll}{x}^2& ,x<0\\ e^{2x}& ,x\ge 0\end{array}$
$f_4:R\to [0,\infty )$
$f_4(x)=\{\begin{array}{ll}{f}_2\left(f_1(x)\right)& ,x<0\\ f_2\left(f_1(x)\right)-1& ,x\ge 0\end{array}$
$=\{\begin{array}{llc}{x}^2& ,& x<0\\ e^{2x}-1& ,& x\ge 0\end{array}$