Property of $\frac{\int_t^\infty x f(x) dx}{\int_t^\infty f(x) dx}$

Question

$f(x)$ is a probability density function over $(0, \infty)$.

Is there any studies on the following function of $t$?

$$g(t)=\frac{\int_t^\infty x f(x) dx}{\int_t^\infty f(x) dx}$$

Thanks.