Let $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}}-\mathrm{x}$ and $\mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2-\mathrm{x}, \forall \mathrm{x}\in \mathrm{R}.$ Then the set of al $\mathrm{x}\in \mathrm{R}$, where the function $\mathrm{h}\left(\mathrm{x}\right)=\left(\text{fog}\right)\left(\mathrm{x}\right)$ is increasing, is:

$$\begin{gathered} f\left(x\right)=e^x-x \\ g\left(x\right)=x^2-x \\ h\left(x\right)=fog\left(x\right) \\ =f\left(g\right(x\left)\right) \\ =f(x^2-x) \\ =e^{x^2-x}-(x^2-x) \\ h\text{'}\left(x\right)=e^{x^2-x}(2x-1)-(2x-1) \\ =(2x-1)(e^{x^2-x}-1) \\ h\left(x\right) is increa\sin g \Rightarrow h\text{'}\left(x\right)>0 \\ (2x-1)(e^{x^2-x}-1)>0 \\ case 1: \\ 2x-1>0 e^{x^2-x}<1 \\ x<\frac{1}{2} x^2-x<0 \\ x\in (0,\frac{1}{2}) \\ x\in (0,\frac{1}{2})\cup (1,\infty ) \end{gathered}$$