If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$ Here $[.]$ is for greatest integer function.
Attempt: since domain of $f(x)$ is $[-1,2]$ therefore for $f([x]-x^2+4)$
$\Rightarrow x^2\le[x]+5$ and $x^2\ge[x]+2$
solving first inequality, as $x^2$ is always positive so $x\ge-5$
Now I can start taking intervals of $x$ and solve them but this brute force method is not taking me anywhere near to the correct answer. Can someone explain me how is this problem solved? Please give an elaborate solution.