1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f1: R arrow R , f2:[0, ∞) arrow R, f3: R arrow R and f4: R arrow[0, ∞) be defined by f1(x)= begincases|x| text if x<0 ex text if x ≥ 0 endcases ; f2(x)=x2 ; f3(x)= begincases sin x text if x<0 x text if x ≥ 0 endcases and f4(x)= begincasesf2(f1(x)) text if x<0 f2(f1(x))-1 text if x ≥ 0 endcases List I List II P f4 is 1 onto but not one-one Q f3 is 2 neither continuous nor one-one R f2 o f1 is 3 differentiable but not one-one S f2 is 4 continuous and one-one

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Q. Let $f_{1}: R \rightarrow R , f_{2}:[0, \infty) \rightarrow R, f_{3}: R \rightarrow R$ and $f_{4}: R \rightarrow[0, \infty)$ be defined by
$f_{1}(x)=\begin{cases}|x| & \text { if } & x<0 \\ e^{x} & \text { if } & x \geq 0\end{cases} ; f_{2}(x)=x^{2} ; f_{3}(x)=\begin{cases}\sin x & \text { if } & x<0 \\ x & \text { if } & x \geq 0\end{cases}$ and $f_{4}(x)=\begin{cases}f_{2}\left(f_{1}(x)\right) & \text { if } & x<0 \\ f_{2}\left(f_{1}(x)\right)-1 & \text { if } & x \geq 0\end{cases}$
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 $f_2 o f_1$ is 3 differentiable but not one-one
S $f_2$ is 4 continuous and one-one

JEE AdvancedJEE Advanced 2014

Solution:

$f_{2}\left(f_{1}\right)=\begin{cases}x^{2} & , x<0 \\ e^{2 x} & , \quad x \geq 0\end{cases}$
$f _{4}: R \rightarrow[0, \infty)$
$f_{4}(x)=\begin{cases}f_{2}\left(f_{1}(x)\right) & , x<0 \\ f_{2}\left(f_{1}(x)\right)-1 & , x \geq 0\end{cases}$
$=\begin{cases}x^{2} & , & x<0 \\ e^{2 x}-1 & , & x \geq 0\end{cases}$
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