jmc

We need $-6x^2+11x-4\ge 0$. The quadratic factors as $$(2x-1)(-3x+4)\ge 0.$$ Thus the zeroes of the quadratic are at $\frac{1}{2}$ and $\frac{4}{3}$. Since the quadratic faces downward, it is nonnegative between the zeroes. So the domain is $x\in \boxed{\left[\frac{1}{2},\frac{4}{3}\right]}$.