Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find all $x \ge 0$ such that $f(x) = 5.$

Question

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$

Find all $x \ge 0$ such that $f(x) = 5.$

I found that $f(2)=4$ and $f(3)=9$ so $f(x)=5$ should be in $x \in (2,3).$ But I found that $f(x)=4$ also works for this interval, meaning my interval is incorrect. Could someone help me out here? Thanks!

Answer 1

If $x < 2.5$, then $x \lfloor x \rfloor < 5$ so $\lfloor x \lfloor x \rfloor \rfloor < 5$.

If $x \geq 3$, then $x \lfloor x \rfloor \geq 9$ so $\lfloor x \lfloor x \rfloor \rfloor \geq 9$.

If $2.5 \leq x < 3$, then $\lfloor x \rfloor = 2$, so $5 \leq x \lfloor x \rfloor < 6$, so $\lfloor x \lfloor x \rfloor \rfloor = 5$.

So the interval is $[2.5, 3)$.