Answer :
To graph the function [tex]\( g(x) \)[/tex] defined as a piecewise function, we will plot each segment of the function across its respective interval. The function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[ g(x) = \left\{ \begin{array}{cc} -2 & \text{if } -1 \leq x < 0 \\ -1 & \text{if } 0 \leq x < 1 \\ 0 & \text{if } 1 \leq x < 2 \\ 1 & \text{if } 2 \leq x < 3 \end{array} \right. \][/tex]
Let's graph [tex]\( g(x) \)[/tex] step-by-step:
1. Interval [tex]\([-1, 0)\)[/tex] with [tex]\( g(x) = -2 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([-1, 0)\)[/tex], [tex]\( g(x) = -2 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = -2 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex] (excluding [tex]\( x = 0 \)[/tex]).
2. Interval [tex]\([0, 1)\)[/tex] with [tex]\( g(x) = -1 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([0, 1)\)[/tex], [tex]\( g(x) = -1 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = -1 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex] (excluding [tex]\( x = 1 \)[/tex]).
3. Interval [tex]\([1, 2)\)[/tex] with [tex]\( g(x) = 0 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([1, 2)\)[/tex], [tex]\( g(x) = 0 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = 0 \)[/tex] from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex] (excluding [tex]\( x = 2 \)[/tex]).
4. Interval [tex]\([2, 3)\)[/tex] with [tex]\( g(x) = 1 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([2, 3)\)[/tex], [tex]\( g(x) = 1 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = 1 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex] (excluding [tex]\( x = 3 \)[/tex]).
To combine all these intervals into a single graph, we plot the horizontal lines specified over their corresponding intervals and include open or closed endpoints where appropriate.
The points used to plot the graph are as follows:
- From [tex]\(-1\)[/tex] to [tex]\(0\)[/tex]: A horizontal line at [tex]\( y = -2 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(-1\)[/tex] to just before [tex]\(0\)[/tex].
- From [tex]\(0\)[/tex] to [tex]\(1\)[/tex]: A horizontal line at [tex]\( y = -1 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(0\)[/tex] to just before [tex]\(1\)[/tex].
- From [tex]\(1\)[/tex] to [tex]\(2\)[/tex]: A horizontal line at [tex]\( y = 0 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(1\)[/tex] to just before [tex]\(2\)[/tex].
- From [tex]\(2\)[/tex] to [tex]\(3\)[/tex]: A horizontal line at [tex]\( y = 1 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(2\)[/tex] to just before [tex]\(3\)[/tex].
Here is a summary of the specific [tex]\( (x, y) \)[/tex] pairs for each interval:
- [tex]\( x \)[/tex] values from [tex]\([-1, -0.002004008016032064, ..., -0.004008016032064243, -0.002004008016032177]\)[/tex] corresponding to [tex]\( y = -2 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([0.0, 0.002004008016032064, ..., 0.9989959919839679, 1.0]\)[/tex] corresponding to [tex]\( y = -1 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([1.0, 1.002004008016032, ..., 1.9989959919839679, 2.0]\)[/tex] corresponding to [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([2.0, 2.002004008016032, ..., 2.9989959919839679, 3.0]\)[/tex] corresponding to [tex]\( y = 1 \)[/tex].
This ensures that the graph covers all segments defined by the piecewise function [tex]\( g(x) \)[/tex], demonstrating each interval's flat, horizontal line characteristic and showcasing the transition of the function's value across different ranges of [tex]\( x \)[/tex].
[tex]\[ g(x) = \left\{ \begin{array}{cc} -2 & \text{if } -1 \leq x < 0 \\ -1 & \text{if } 0 \leq x < 1 \\ 0 & \text{if } 1 \leq x < 2 \\ 1 & \text{if } 2 \leq x < 3 \end{array} \right. \][/tex]
Let's graph [tex]\( g(x) \)[/tex] step-by-step:
1. Interval [tex]\([-1, 0)\)[/tex] with [tex]\( g(x) = -2 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([-1, 0)\)[/tex], [tex]\( g(x) = -2 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = -2 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex] (excluding [tex]\( x = 0 \)[/tex]).
2. Interval [tex]\([0, 1)\)[/tex] with [tex]\( g(x) = -1 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([0, 1)\)[/tex], [tex]\( g(x) = -1 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = -1 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex] (excluding [tex]\( x = 1 \)[/tex]).
3. Interval [tex]\([1, 2)\)[/tex] with [tex]\( g(x) = 0 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([1, 2)\)[/tex], [tex]\( g(x) = 0 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = 0 \)[/tex] from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex] (excluding [tex]\( x = 2 \)[/tex]).
4. Interval [tex]\([2, 3)\)[/tex] with [tex]\( g(x) = 1 \)[/tex]:
- For [tex]\( x \)[/tex] in the range [tex]\([2, 3)\)[/tex], [tex]\( g(x) = 1 \)[/tex].
- The graph should show a horizontal line at [tex]\( y = 1 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex] (excluding [tex]\( x = 3 \)[/tex]).
To combine all these intervals into a single graph, we plot the horizontal lines specified over their corresponding intervals and include open or closed endpoints where appropriate.
The points used to plot the graph are as follows:
- From [tex]\(-1\)[/tex] to [tex]\(0\)[/tex]: A horizontal line at [tex]\( y = -2 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(-1\)[/tex] to just before [tex]\(0\)[/tex].
- From [tex]\(0\)[/tex] to [tex]\(1\)[/tex]: A horizontal line at [tex]\( y = -1 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(0\)[/tex] to just before [tex]\(1\)[/tex].
- From [tex]\(1\)[/tex] to [tex]\(2\)[/tex]: A horizontal line at [tex]\( y = 0 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(1\)[/tex] to just before [tex]\(2\)[/tex].
- From [tex]\(2\)[/tex] to [tex]\(3\)[/tex]: A horizontal line at [tex]\( y = 1 \)[/tex] for [tex]\( x \)[/tex] values from [tex]\(2\)[/tex] to just before [tex]\(3\)[/tex].
Here is a summary of the specific [tex]\( (x, y) \)[/tex] pairs for each interval:
- [tex]\( x \)[/tex] values from [tex]\([-1, -0.002004008016032064, ..., -0.004008016032064243, -0.002004008016032177]\)[/tex] corresponding to [tex]\( y = -2 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([0.0, 0.002004008016032064, ..., 0.9989959919839679, 1.0]\)[/tex] corresponding to [tex]\( y = -1 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([1.0, 1.002004008016032, ..., 1.9989959919839679, 2.0]\)[/tex] corresponding to [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex] values from [tex]\([2.0, 2.002004008016032, ..., 2.9989959919839679, 3.0]\)[/tex] corresponding to [tex]\( y = 1 \)[/tex].
This ensures that the graph covers all segments defined by the piecewise function [tex]\( g(x) \)[/tex], demonstrating each interval's flat, horizontal line characteristic and showcasing the transition of the function's value across different ranges of [tex]\( x \)[/tex].
