Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
