Let \(\mathrm{f}(x)=\mathrm{e}^x-x\) and \(\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}\), then the set of all \(x \in \mathrm{R}\), where the function \(\mathrm{h}(x)=(\mathrm{fog})(x)\) is increasing is

\(\begin{aligned}
& \mathrm{h}(x)=(\mathrm{fog})(x) \\
& \Rightarrow \mathrm{h}(x)=\mathrm{f}\left(x^2-x\right) \\
& \Rightarrow \mathrm{h}(x)=\mathrm{e}^{x^2-x}-x^2+x \\
& \therefore \quad \mathrm{h}^{\prime}(x)=\mathrm{e}^{x^2-x}(2 x-1)-2 x+1 \\
& \Rightarrow \mathrm{h}^{\prime}(x)=\left(\mathrm{e}^{x^2-x}-1\right)(2 x-1)
\end{aligned}\)
For function \(\mathrm{h}(x)\) to be increasing,
\(\begin{aligned}
& \mathrm{h}^{\prime}(x) \geq 0 \\
& \Rightarrow\left(\mathrm{e}^{x^2-x}-1\right)(2 x-1) \geq 0 \\
& \Rightarrow x \in\left[0, \frac{1}{2}\right] \cup[1, \infty)
\end{aligned}\)
\(\begin{array}{c|c|c|c}
\mathrm{h}^{\prime}- & + & - & + \\
\hline & & & \\
0 & \frac{1}{2} & 1
\end{array}\)