Answer :
Let's analyze the piecewise function [tex]\( f(x) \)[/tex] to understand its behavior across different intervals. The function is defined as follows:
[tex]\[ f(x)=\left\{ \begin{array}{ll} 5, & x < -2 \\ 3, & -2 \leq x < 0 \\ 0, & 0 \leq x < 2 \\ -3, & x \geq 2 \end{array} \right. \][/tex]
To sketch and interpret the graph of this function, we'll examine it piece by piece:
1. For [tex]\( x < -2 \)[/tex]:
- In this interval, [tex]\( f(x) = 5 \)[/tex]. This means the function value is constant and equal to 5 for all [tex]\( x \)[/tex] less than -2.
- Graphically, this would be a horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
2. For [tex]\( -2 \leq x < 0 \)[/tex]:
- In this range, [tex]\( f(x) = 3 \)[/tex]. Here, the function value remains constant and equal to 3 for [tex]\( x \)[/tex] between -2 and 0 (inclusive of -2, exclusive of 0).
- On the graph, this appears as a horizontal line at [tex]\( y = 3 \)[/tex], starting from [tex]\( x = -2 \)[/tex] (inclusive) to just before [tex]\( x = 0 \)[/tex].
3. For [tex]\( 0 \leq x < 2 \)[/tex]:
- Within this interval, [tex]\( f(x) = 0 \)[/tex]. The function value is constant and zero.
- This is shown on the graph as a horizontal line at [tex]\( y = 0 \)[/tex] extending from [tex]\( x = 0 \)[/tex] (inclusive) to just before [tex]\( x = 2 \)[/tex].
4. For [tex]\( x \geq 2 \)[/tex]:
- Here, [tex]\( f(x) = -3 \)[/tex]. The function value is constant and equal to -3 for all [tex]\( x \)[/tex] values greater than or equal to 2.
- This is depicted as a horizontal line at [tex]\( y = -3 \)[/tex] starting from [tex]\( x = 2 \)[/tex] and extending indefinitely for all [tex]\( x > 2 \)[/tex].
### Identification of Key Points
- The function changes values at points [tex]\( x = -2 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex]. Let's clarify the behavior at these points:
- At [tex]\( x = -2 \)[/tex], the function value transitions from 5 (for [tex]\( x < -2 \)[/tex]) to 3 (for [tex]\( x \geq -2 \)[/tex]).
- At [tex]\( x = 0 \)[/tex], the value shifts from 3 (for [tex]\( x < 0 \)[/tex]) to 0 (for [tex]\( x \geq 0 \)[/tex]).
- At [tex]\( x = 2 \)[/tex], the function changes from 0 (for [tex]\( x < 2 \)[/tex]) to -3 (for [tex]\( x \geq 2 \)[/tex]).
### Graph Characteristics
- The graph will consist of horizontal line segments at the specified function values within their respective intervals.
- There will be closed circles (solid dots) at the endpoint where the function value is included.
- There will be open circles (hollow dots) at the endpoint where the function value is not included.
For example:
- At [tex]\( x = -2 \)[/tex], there should be a solid dot at [tex]\( y = 3 \)[/tex] and an open dot at [tex]\( y = 5 \)[/tex].
- At [tex]\( x = 0 \)[/tex], an open dot at [tex]\( y = 3 \)[/tex] and a solid dot at [tex]\( y = 0 \)[/tex].
- At [tex]\( x = 2 \)[/tex], an open dot at [tex]\( y = 0 \)[/tex] and a solid dot at [tex]\( y = -3 \)[/tex].
Given these segments and points, you can visualize and identify the correct graph representing the function [tex]\( f(x) \)[/tex]. The function should:
- Be a horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
- Transition to a horizontal line at [tex]\( y = 3 \)[/tex] for [tex]\( -2 \leq x < 0 \)[/tex].
- Then transition to a horizontal line at [tex]\( y = 0 \)[/tex] for [tex]\( 0 \leq x < 2 \)[/tex].
- Finally, be a horizontal line at [tex]\( y = -3 \)[/tex] for [tex]\( x \geq 2 \)[/tex].
Thus, by piecing together these segments and considering the correct inclusion and exclusion of endpoints using open and closed circles, you can accurately sketch the graph representing this piecewise function.
[tex]\[ f(x)=\left\{ \begin{array}{ll} 5, & x < -2 \\ 3, & -2 \leq x < 0 \\ 0, & 0 \leq x < 2 \\ -3, & x \geq 2 \end{array} \right. \][/tex]
To sketch and interpret the graph of this function, we'll examine it piece by piece:
1. For [tex]\( x < -2 \)[/tex]:
- In this interval, [tex]\( f(x) = 5 \)[/tex]. This means the function value is constant and equal to 5 for all [tex]\( x \)[/tex] less than -2.
- Graphically, this would be a horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
2. For [tex]\( -2 \leq x < 0 \)[/tex]:
- In this range, [tex]\( f(x) = 3 \)[/tex]. Here, the function value remains constant and equal to 3 for [tex]\( x \)[/tex] between -2 and 0 (inclusive of -2, exclusive of 0).
- On the graph, this appears as a horizontal line at [tex]\( y = 3 \)[/tex], starting from [tex]\( x = -2 \)[/tex] (inclusive) to just before [tex]\( x = 0 \)[/tex].
3. For [tex]\( 0 \leq x < 2 \)[/tex]:
- Within this interval, [tex]\( f(x) = 0 \)[/tex]. The function value is constant and zero.
- This is shown on the graph as a horizontal line at [tex]\( y = 0 \)[/tex] extending from [tex]\( x = 0 \)[/tex] (inclusive) to just before [tex]\( x = 2 \)[/tex].
4. For [tex]\( x \geq 2 \)[/tex]:
- Here, [tex]\( f(x) = -3 \)[/tex]. The function value is constant and equal to -3 for all [tex]\( x \)[/tex] values greater than or equal to 2.
- This is depicted as a horizontal line at [tex]\( y = -3 \)[/tex] starting from [tex]\( x = 2 \)[/tex] and extending indefinitely for all [tex]\( x > 2 \)[/tex].
### Identification of Key Points
- The function changes values at points [tex]\( x = -2 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex]. Let's clarify the behavior at these points:
- At [tex]\( x = -2 \)[/tex], the function value transitions from 5 (for [tex]\( x < -2 \)[/tex]) to 3 (for [tex]\( x \geq -2 \)[/tex]).
- At [tex]\( x = 0 \)[/tex], the value shifts from 3 (for [tex]\( x < 0 \)[/tex]) to 0 (for [tex]\( x \geq 0 \)[/tex]).
- At [tex]\( x = 2 \)[/tex], the function changes from 0 (for [tex]\( x < 2 \)[/tex]) to -3 (for [tex]\( x \geq 2 \)[/tex]).
### Graph Characteristics
- The graph will consist of horizontal line segments at the specified function values within their respective intervals.
- There will be closed circles (solid dots) at the endpoint where the function value is included.
- There will be open circles (hollow dots) at the endpoint where the function value is not included.
For example:
- At [tex]\( x = -2 \)[/tex], there should be a solid dot at [tex]\( y = 3 \)[/tex] and an open dot at [tex]\( y = 5 \)[/tex].
- At [tex]\( x = 0 \)[/tex], an open dot at [tex]\( y = 3 \)[/tex] and a solid dot at [tex]\( y = 0 \)[/tex].
- At [tex]\( x = 2 \)[/tex], an open dot at [tex]\( y = 0 \)[/tex] and a solid dot at [tex]\( y = -3 \)[/tex].
Given these segments and points, you can visualize and identify the correct graph representing the function [tex]\( f(x) \)[/tex]. The function should:
- Be a horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
- Transition to a horizontal line at [tex]\( y = 3 \)[/tex] for [tex]\( -2 \leq x < 0 \)[/tex].
- Then transition to a horizontal line at [tex]\( y = 0 \)[/tex] for [tex]\( 0 \leq x < 2 \)[/tex].
- Finally, be a horizontal line at [tex]\( y = -3 \)[/tex] for [tex]\( x \geq 2 \)[/tex].
Thus, by piecing together these segments and considering the correct inclusion and exclusion of endpoints using open and closed circles, you can accurately sketch the graph representing this piecewise function.
