Sketch the function $\displaystyle f(x)=\min_{t\le x} t^2$ for all real $x$.

What is the minimum value of $t^2$ when $t$ varies between $-\infty$ and $-3?$

How about between $-\infty$ and $-1?$

How about between $-\infty$ and $+1?$

We are looking for the minimum value of $t^2$ for $t\in(-\infty,x]$. This is $x^2$ when $x<0$, and $0$ when $x\ge0$.