Question

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 $f_1\left(x\right)=\{\begin{array}{c}|x|,ifx<0\\ e^x,ifx\ge 0\end{array}{f}_2\left(x\right)=x^2;f_3\left(x\right)=\{\begin{array}{c}sinx,ifx<0\\ x,ifx\ge 0\end{array}$ and $f_4\left(x\right)=\{\begin{array}{c}{f}_2\left[f_1\right(x\left)\right],ifx<0\\ f_2\left[f_1\right(x\left)\right]-1,ifx\ge 0\end{array}$
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ a.f_{4}is& p.\text{onto but not one-one}\\ b.f_{3}is& q.\text{neither continuous nor one-one}\\ c.f_{2}of{f}_{1}is& r.\text{differentiable but not one-one}\\ d.f_{2}is& s.\text{continuous and one-one}\end{array}$

• A. A-r B-p C-s D-q
• B. A-p B-r C-s D-q
• C. A-r B-p C-q D-s
• D. A-p B-r C-q D-s

Answer 1

The correct option is D

A-p B-r C-q D-s

$f_1\left(x\right)=\{\begin{array}{c}-x,x<0\\ e^x,x\ge 0\end{array}$
$f_2\left(x\right)=x^2,x\ge 0$
$f_3\left(x\right)=\{\begin{array}{c}sinx,x<0\\ x,x\ge 0\end{array}$
$f_4\left(x\right)=\{\begin{array}{c}{f}_2\left(f_1\right(x\left)\right),x<0\\ f_2\left(f_{(}x\right))-1,x\ge 0\end{array}$
Now, $f_2\left(f_1\right(x\left)\right)=\{\begin{array}{c}{x}^2,x<0\\ e^{2x},x\ge 0\end{array}$
$\Rightarrow f_4=\{\begin{array}{c}{x}^2,x<0\\ e^{2x}-1,x\ge 0\end{array}$
As $f_4\left(x\right)$ is continuous, $f_4^{\prime }\left(x\right)=\{\begin{array}{c}2x,x<0\\ 2e^{2x},x>0\end{array}$

$f{;}_4\left(0\right)$ is not defined. Its range is $[0,\infty )$.
Thus, range = coadmin = $[0,\infty )$, thus $f_4$ is onto.
Also, horizontal line(drawn parallel to X-axis) meets the curve more than once, thus function is not one-one.