$\begin{aligned}
& f(x)=x-1 \
& f(f(x))=f(x)-1=x-1-1=x-2 \
& g(f(f(x)))=e^{x-2} \
& \therefore \frac{d y}{d x}=e^{-2 \sqrt{x}} \times e^{x-2}-\frac{1}{\sqrt{x}} y \
& \frac{d y}{d x}+\frac{1}{\sqrt{x}} y=e^{x-2 \sqrt{x}-2} \text { which is L.D.E } \
& \text { I.F. }=e^{\int \frac{d y}{\sqrt{x}}}=e^{2 \sqrt{x}}
\end{aligned}$
Its solution is
$\begin{aligned}
& y \times e^{2 \sqrt{x}}=\int e^{2 \sqrt{x}} \times e^{x-2 \sqrt{x}-2} d x+c \
& y \times e^{2 \sqrt{x}}=\int e^{x-2} d x+c \
& y \times e^{2 \sqrt{x}}=e^{x-2}+c
\end{aligned}$
Given x=0,y=0⇒0=e−2+c;c=−e−2
∴y×e2x=ex−2−e−2
when x=1,y×e2=e−1−e−2
y=e2e−1−e−2=e2e1−e21=e5e2−e=e4e−1
Option (1) is correct