I have following function $$f(x)=-\operatorname{Ei}\left(-a-\frac{b}{x}\right) \exp\left(a+\frac{b}{x}\right)$$ where $\operatorname{Ei}(x)$ is the exponential integral and $a>0$ and $b>0$. If I assume that $a+\frac{b}{x}=\frac{1}{t}$ then it becomes $$g(t)=-\operatorname{Ei} \left(-\frac{1}{t}\right)\exp\left(\frac{1}{t} \right)$$ and when I plot $g(t)$ is WA it appears to be a concave function (it plot looks like but I am not sure). I have following two questions.
1- If $g(t)$ is concave then can I say that $f(x)$ is also concave?
2- Is $g(t)$ concave function?
I will be very thankful to you for your help.