Answer :
To graph the piecewise function [tex]\( f(x) = \begin{cases}
-1.5x + 3.5, & x < 2 \\
4 + x, & x \geq 2
\end{cases} \)[/tex], we need to handle each part of the function separately and then combine them on the same set of axes.
### Step-by-Step Solution:
1. Graph the first piece:
- This is the function [tex]\( f(x) = -1.5x + 3.5 \)[/tex] for [tex]\( x < 2 \)[/tex].
- This is a linear function with a slope of -1.5 and a y-intercept of 3.5.
- To graph this, you can plot the y-intercept (0, 3.5) and use the slope to find another point. For example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 3.5 \)[/tex].
- Another point: if [tex]\( x = 2 \)[/tex], then [tex]\( y = -1.5(2) + 3.5 = 0.5 \)[/tex].
- But remember, this part of the function stops at [tex]\( x < 2 \)[/tex], so it does not include [tex]\( x = 2 \)[/tex].
2. Graph the second piece:
- This is the function [tex]\( f(x) = 4 + x \)[/tex] for [tex]\( x \geq 2 \)[/tex].
- This is a linear function with a slope of 1 and a y-intercept of 4.
- To graph this, use:
- The point from the piecewise condition when [tex]\( x = 2 \)[/tex], thus [tex]\( y = 4 + 2 = 6 \)[/tex].
- Another point: if [tex]\( x = 3 \)[/tex], then [tex]\( y = 4 + 3 = 7 \)[/tex].
3. Combine both pieces:
- Plot both linear segments on the same set of axes.
- For [tex]\( x < 2 \)[/tex], you have the line segment from the function [tex]\( -1.5x + 3.5 \)[/tex] heading towards [tex]\( x = 2 \)[/tex].
- At [tex]\( x = 2 \)[/tex], check the continuity:
- [tex]\( y \)[/tex] should transition smoothly between the pieces.
- For [tex]\( x = 2 \)[/tex], the value of the first piece at the boundary is [tex]\( -1.5(2) + 3.5 = 0.5 \)[/tex].
- The second piece starts at [tex]\( (2, 6) \)[/tex].
4. Plot specific points and the resulting lines:
- Points on the first segment (for [tex]\( x < 2 \)[/tex]):
- Start at [tex]\( x = 0, y = 3.5 \)[/tex]
- Point just before [tex]\( x = 2 \)[/tex] (e.g., [tex]\( x = 1.5 \)[/tex], calculate [tex]\( y = -1.5(1.5) + 3.5 = 1.75 \)[/tex]).
- Ends just before [tex]\( x = 2 \)[/tex], open circle at [tex]\( (2, 0.5) \)[/tex].
- Points on the second segment (for [tex]\( x \geq 2 \)[/tex]):
- Start at [tex]\( (2, 6) \)[/tex] and include it with a closed circle.
- As [tex]\( x \)[/tex] increases to 3, [tex]\( y = 7 \)[/tex].
5. Graph construction:
- Draw two segments based on points calculated:
- The first line segment from [tex]\( (0, 3.5) \)[/tex] to an open circle at [tex]\( (2, 0.5) \)[/tex].
- The second line segment starts at closed circle [tex]\( (2, 6) \)[/tex] and continues onward through points like [tex]\( (3, 7) \)[/tex].
By following the above steps, you can effectively construct and understand the given piecewise-defined function graph. This graph transitions from one linear segment to another at [tex]\( x = 2 \)[/tex]. The key aspects are ensuring the correct linear behaviors and the proper handling of open and closed circles at the boundary point [tex]\( x = 2 \)[/tex].
### Step-by-Step Solution:
1. Graph the first piece:
- This is the function [tex]\( f(x) = -1.5x + 3.5 \)[/tex] for [tex]\( x < 2 \)[/tex].
- This is a linear function with a slope of -1.5 and a y-intercept of 3.5.
- To graph this, you can plot the y-intercept (0, 3.5) and use the slope to find another point. For example:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 3.5 \)[/tex].
- Another point: if [tex]\( x = 2 \)[/tex], then [tex]\( y = -1.5(2) + 3.5 = 0.5 \)[/tex].
- But remember, this part of the function stops at [tex]\( x < 2 \)[/tex], so it does not include [tex]\( x = 2 \)[/tex].
2. Graph the second piece:
- This is the function [tex]\( f(x) = 4 + x \)[/tex] for [tex]\( x \geq 2 \)[/tex].
- This is a linear function with a slope of 1 and a y-intercept of 4.
- To graph this, use:
- The point from the piecewise condition when [tex]\( x = 2 \)[/tex], thus [tex]\( y = 4 + 2 = 6 \)[/tex].
- Another point: if [tex]\( x = 3 \)[/tex], then [tex]\( y = 4 + 3 = 7 \)[/tex].
3. Combine both pieces:
- Plot both linear segments on the same set of axes.
- For [tex]\( x < 2 \)[/tex], you have the line segment from the function [tex]\( -1.5x + 3.5 \)[/tex] heading towards [tex]\( x = 2 \)[/tex].
- At [tex]\( x = 2 \)[/tex], check the continuity:
- [tex]\( y \)[/tex] should transition smoothly between the pieces.
- For [tex]\( x = 2 \)[/tex], the value of the first piece at the boundary is [tex]\( -1.5(2) + 3.5 = 0.5 \)[/tex].
- The second piece starts at [tex]\( (2, 6) \)[/tex].
4. Plot specific points and the resulting lines:
- Points on the first segment (for [tex]\( x < 2 \)[/tex]):
- Start at [tex]\( x = 0, y = 3.5 \)[/tex]
- Point just before [tex]\( x = 2 \)[/tex] (e.g., [tex]\( x = 1.5 \)[/tex], calculate [tex]\( y = -1.5(1.5) + 3.5 = 1.75 \)[/tex]).
- Ends just before [tex]\( x = 2 \)[/tex], open circle at [tex]\( (2, 0.5) \)[/tex].
- Points on the second segment (for [tex]\( x \geq 2 \)[/tex]):
- Start at [tex]\( (2, 6) \)[/tex] and include it with a closed circle.
- As [tex]\( x \)[/tex] increases to 3, [tex]\( y = 7 \)[/tex].
5. Graph construction:
- Draw two segments based on points calculated:
- The first line segment from [tex]\( (0, 3.5) \)[/tex] to an open circle at [tex]\( (2, 0.5) \)[/tex].
- The second line segment starts at closed circle [tex]\( (2, 6) \)[/tex] and continues onward through points like [tex]\( (3, 7) \)[/tex].
By following the above steps, you can effectively construct and understand the given piecewise-defined function graph. This graph transitions from one linear segment to another at [tex]\( x = 2 \)[/tex]. The key aspects are ensuring the correct linear behaviors and the proper handling of open and closed circles at the boundary point [tex]\( x = 2 \)[/tex].
