graph each piece - wise function.\n1. $f(x)=\begin{cases}2x + 3&-3<x<0\\3&xgeq0end{cases}$\n2…

Explanation:

Step1: Analyze first - piece of function 1

For (f(x)=2x + 3), when (x=-3), (f(-3)=2\times(-3)+3=-3); when (x = 0), (f(0)=2\times0+3 = 3). Plot the line segment for (-3<x<0).

Step2: Analyze second - piece of function 1

For (f(x)=3) when (x\geq0), it is a horizontal line at (y = 3) starting from (x = 0) (including (x = 0)).

Step3: Analyze first - piece of function 2

For (f(x)=\frac{1}{2}x+2) when (x\leq - 2), when (x=-2), (f(-2)=\frac{1}{2}\times(-2)+2=1). Plot the line for (x\leq - 2).

Step4: Analyze second - piece of function 2

For (f(x)=-x) when (x>-2), when (x=-2), (f(-2)=-(-2)=2). Plot the line for (x > - 2).

Step5: Analyze first - piece of function 3

For (f(x)=1) when (x\leq1), it is a horizontal line at (y = 1) for (x\leq1).

Step6: Analyze second - piece of function 3

For (f(x)=-1) when (x>2), it is a horizontal line at (y=-1) for (x>2).

Step7: Analyze first - piece of function 4

For (f(x)=\frac{2}{3}x + 4) when (x\leq - 3), when (x=-3), (f(-3)=\frac{2}{3}\times(-3)+4=2). Plot the line for (x\leq - 3).

Step8: Analyze second - piece of function 4

For (f(x)=2) when (-3<x<1), it is a horizontal line at (y = 2) for (-3<x<1).

Step9: Analyze third - piece of function 4

For (f(x)=2x) when (x>1), when (x = 1), (f(1)=2\times1=2). Plot the line for (x>1).

Step10: Analyze first - piece of function 5

For (f(x)=3x + 6) when (x\leq - 1), when (x=-1), (f(-1)=3\times(-1)+6=3). Plot the line for (x\leq - 1).

Step11: Analyze second - piece of function 5

For (f(x)=-x + 2) when (-1<x<1), when (x=-1), (f(-1)=-(-1)+2=3); when (x = 1), (f(1)=-1 + 2=1). Plot the line segment for (-1<x<1).

Step12: Analyze third - piece of function 5

For (f(x)=1) when (x\geq1), it is a horizontal line at (y = 1) for (x\geq1).

Step13: Analyze first - piece of function 6

For (f(x)=2x+7) when (x\leq - 3), when (x=-3), (f(-3)=2\times(-3)+7=1). Plot the line for (x\leq - 3).

Step14: Analyze second - piece of function 6

For (f(x)=\frac{1}{3}x+2) when (x>-3), when (x=-3), (f(-3)=\frac{1}{3}\times(-3)+2=1). Plot the line for (x>-3).

To graph each piece - wise function, follow the above - described steps for plotting lines and line - segments according to the given domain for each sub - function. The actual graph cannot be directly presented in text but the steps to create it are provided.

Since the problem is about graphing piece - wise functions which are a part of coordinate geometry, the general approach is to find key points (end - points of the domain of each sub - function) and then plot the lines or line - segments accordingly.

Answer:

The steps to graph each of the given piece - wise functions are provided above. For a complete solution, use graph paper or a graphing utility to plot the lines and line - segments as described in each step for each function.