At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To graph the function [tex]\( h \)[/tex] defined as:
[tex]\[ h(x) = \begin{cases} x & \text{if } x \neq 2 \\ 3 & \text{if } x = 2 \end{cases} \][/tex]
we will proceed through the following steps:
1. Understand the Function:
- For all values of [tex]\( x \)[/tex] except [tex]\( x = 2 \)[/tex], the function [tex]\( h(x) \)[/tex] is linear and defined as [tex]\( h(x) = x \)[/tex].
- Exactly at [tex]\( x = 2 \)[/tex], the function [tex]\( h(x) \)[/tex] takes the value [tex]\( h(2) = 3 \)[/tex], which is a point discontinuity (it differs from the value [tex]\( h(x) = x \)[/tex] that would normally be [tex]\( h(2) = 2 \)[/tex]).
2. Graph the Linear Part:
- Plot the line [tex]\( y = x \)[/tex] for all [tex]\( x \neq 2 \)[/tex]. This line passes through the origin [tex]\((0, 0)\)[/tex] and has a slope of 1 (i.e., for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] also increases by 1).
3. Mark the Point of Discontinuity:
- At [tex]\( x = 2 \)[/tex], instead of following the line [tex]\( y = x \)[/tex], the function [tex]\( h(x) \)[/tex] jumps to [tex]\( h(2) = 3 \)[/tex].
- Represent this point [tex]\((2, 3)\)[/tex] with a filled circle to indicate that the function takes this value at [tex]\( x = 2 \)[/tex].
- Draw an open circle at the point [tex]\((2, 2)\)[/tex] on the line [tex]\( y = x \)[/tex] to illustrate that the value [tex]\( x = 2 \)[/tex] is not included in the linear part of the function.
4. Add Axes and Labels:
- Draw and label the [tex]\( x \)[/tex]-axis and [tex]\( y \)[/tex]-axis.
- Provide axis labels and appropriate tick marks for reference.
### Detailed Steps to Plot:
1. Draw the Line [tex]\( y = x \)[/tex]:
- For [tex]\( x \)[/tex] ranging from a wide interval like [tex]\([-10, 10]\)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -10 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 10 \)[/tex]
- Connect these points with a straight line, but exclude the point where [tex]\( x = 2 \)[/tex].
2. Add the Special Points:
- Draw an open circle at [tex]\((2, 2)\)[/tex] to indicate that [tex]\( h(x) \)[/tex] does not equal 2 when [tex]\( x = 2 \)[/tex].
- Put a filled circle at [tex]\((2, 3)\)[/tex] to show that [tex]\( h(2) = 3 \)[/tex].
### Final Graph:
1. Draw the [tex]\( y = x \)[/tex] Line Excluding [tex]\( x = 2 \)[/tex]:
- A diagonal line passing through the origin (0,0) and continuing in both positive and negative directions, with a gap at [tex]\( x = 2 \)[/tex].
2. Mark [tex]\( (2, 3) \)[/tex] with a Filled Circle:
- This distinct point signifies the value of the function at [tex]\( x = 2 \)[/tex].
3. Relevant Labels:
- Label the vertical axis (y-axis) and horizontal axis (x-axis).
- Optionally, label specific points or add grid lines for clarity.
This graph will illustrate both the continuous linear behavior of [tex]\( h \)[/tex] away from [tex]\( x = 2 \)[/tex] and the unique behavior of [tex]\( h \)[/tex] at [tex]\( x = 2 \)[/tex].
[tex]\[ h(x) = \begin{cases} x & \text{if } x \neq 2 \\ 3 & \text{if } x = 2 \end{cases} \][/tex]
we will proceed through the following steps:
1. Understand the Function:
- For all values of [tex]\( x \)[/tex] except [tex]\( x = 2 \)[/tex], the function [tex]\( h(x) \)[/tex] is linear and defined as [tex]\( h(x) = x \)[/tex].
- Exactly at [tex]\( x = 2 \)[/tex], the function [tex]\( h(x) \)[/tex] takes the value [tex]\( h(2) = 3 \)[/tex], which is a point discontinuity (it differs from the value [tex]\( h(x) = x \)[/tex] that would normally be [tex]\( h(2) = 2 \)[/tex]).
2. Graph the Linear Part:
- Plot the line [tex]\( y = x \)[/tex] for all [tex]\( x \neq 2 \)[/tex]. This line passes through the origin [tex]\((0, 0)\)[/tex] and has a slope of 1 (i.e., for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] also increases by 1).
3. Mark the Point of Discontinuity:
- At [tex]\( x = 2 \)[/tex], instead of following the line [tex]\( y = x \)[/tex], the function [tex]\( h(x) \)[/tex] jumps to [tex]\( h(2) = 3 \)[/tex].
- Represent this point [tex]\((2, 3)\)[/tex] with a filled circle to indicate that the function takes this value at [tex]\( x = 2 \)[/tex].
- Draw an open circle at the point [tex]\((2, 2)\)[/tex] on the line [tex]\( y = x \)[/tex] to illustrate that the value [tex]\( x = 2 \)[/tex] is not included in the linear part of the function.
4. Add Axes and Labels:
- Draw and label the [tex]\( x \)[/tex]-axis and [tex]\( y \)[/tex]-axis.
- Provide axis labels and appropriate tick marks for reference.
### Detailed Steps to Plot:
1. Draw the Line [tex]\( y = x \)[/tex]:
- For [tex]\( x \)[/tex] ranging from a wide interval like [tex]\([-10, 10]\)[/tex]:
- When [tex]\( x = -10 \)[/tex], [tex]\( y = -10 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 10 \)[/tex]
- Connect these points with a straight line, but exclude the point where [tex]\( x = 2 \)[/tex].
2. Add the Special Points:
- Draw an open circle at [tex]\((2, 2)\)[/tex] to indicate that [tex]\( h(x) \)[/tex] does not equal 2 when [tex]\( x = 2 \)[/tex].
- Put a filled circle at [tex]\((2, 3)\)[/tex] to show that [tex]\( h(2) = 3 \)[/tex].
### Final Graph:
1. Draw the [tex]\( y = x \)[/tex] Line Excluding [tex]\( x = 2 \)[/tex]:
- A diagonal line passing through the origin (0,0) and continuing in both positive and negative directions, with a gap at [tex]\( x = 2 \)[/tex].
2. Mark [tex]\( (2, 3) \)[/tex] with a Filled Circle:
- This distinct point signifies the value of the function at [tex]\( x = 2 \)[/tex].
3. Relevant Labels:
- Label the vertical axis (y-axis) and horizontal axis (x-axis).
- Optionally, label specific points or add grid lines for clarity.
This graph will illustrate both the continuous linear behavior of [tex]\( h \)[/tex] away from [tex]\( x = 2 \)[/tex] and the unique behavior of [tex]\( h \)[/tex] at [tex]\( x = 2 \)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
