Answer
Please see graph $\\$
(and details in the "work step by step" section
Work Step by Step
We start with $ f(x)=e^{x}\qquad$(red)
... stretch horizontally by factor 2 ,$ f(\displaystyle \frac{1}{2}x)\qquad$(green dashed)
... and then stretch vertically by factor 2, $\displaystyle \quad 2f(\frac{1}{2}x)$
and we obtain $ g(x)=2e^{x/2}=2f(\displaystyle \frac{1}{2}x)\qquad$ (blue)
asymptote of $ f:\quad y=0$
asymptote of $ g:\quad y=0$
domain of $ f:\quad (-\infty,\ \infty)$
range of $ f:\quad (0,\infty)$
domain of $ g:\quad (-\infty,\ \infty)$
range of $ g:\quad (0,\infty)$
