writing step functions\nwrite an expression and interval for each colored region.\nf(x) ={

Explanation:

Step1: Identify bottom - left region

For the bottom - left region with open - circle endpoints, assume the horizontal line is at $y = - 2$. The $x$ - values range from $x=-4$ to $x = - 2$. The expression is $y=-2$ and the interval is $-4\lt x\lt - 2$.

Step2: Identify middle - left region

For the middle - left region with a closed - circle on the left and open - circle on the right, assume the horizontal line is at $y = 1$. The $x$ - values range from $x=-2$ to $x = 0$. The expression is $y = 1$ and the interval is $-2\leqslant x\lt0$.

Step3: Identify right - hand region

For the right - hand region with closed - circle endpoints, assume the horizontal line is at $y=-1$. The $x$ - values range from $x = 0$ to $x = 2$. The expression is $y=-1$ and the interval is $0\leqslant x\leqslant2$.

Answer:

$f(x)=\begin{cases}-2, &-4\lt x\lt - 2\1, &-2\leqslant x\lt0\-1, &0\leqslant x\leqslant2\end{cases}$