3.14: Composite exponential function

It is given that $\mathrm{f}(x)=a+2\mathrm{e}^{x}$ for $x\geqslant 0$, where $a$ is an integer, and $\mathrm{g}(x)=x-1$ for $x\geqslant 1$. Find the least value of $a$ so that the function $\mathrm{gf}$ exists for all $x\geqslant 0$.

In the case where $a=5$, solve $\mathrm{gf}(x)=3$. Give your answer correct to 3 decimal places.