Answer
(a) $f(x) = \lfloor x \rfloor$
(b) $f(x) = \lfloor x^2 \rfloor$
(c) $f(x) = \lfloor x \rfloor^2$
(d) $f(x) = \lfloor \sin x \rfloor$
Work Step by Step
Function $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, creating a step at every integer.
For $\lfloor x^2 \rfloor$, we square $x$ first and then apply the floor, with jumps at $x = \pm \sqrt{k}$ where $k$ is an integer.
For $\lfloor x \rfloor^2$, we take the integer part of $x$ (say $n$) and then square it, so on each interval $[n, n+1)$, the value is $n^2$.
Finally, $\lfloor \sin x \rfloor$ can only be $-1$, $0$, or $1$ becasue $\sin x$ ranges from $-1$ to $1$. It equals $1$ only where $\sin x = 1$, it is $0$ when $0 \le \sin x < 1$, and it is $-1$ when $\sin x < 0$.
