Answer :
Let's start by understanding the provided piecewise function and then move on to graphing it.
The function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[ f(x) = \begin{cases} -x + 4 & \text{if } x < 1 \\ 2x + 1 & \text{if } x \geq 1 \end{cases} \][/tex]
### Step-by-Step Solution:
1. Graphing the Piecewise Functions:
a. For [tex]\( x < 1 \)[/tex]:
The function is [tex]\( f(x) = -x + 4 \)[/tex].
- This is a linear function with a slope of -1 and a y-intercept of 4.
- To graph this, plot the point where [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -0 + 4 = 4 \)[/tex].
- Because it’s a line, select another point to better draw the line. For example, choose [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = -(-1) + 4 = 5 \)[/tex].
So, the line for [tex]\( x < 1 \)[/tex] passes through the points (0, 4) and (-1, 5).
b. For [tex]\( x \geq 1 \)[/tex]:
The function is [tex]\( f(x) = 2x + 1 \)[/tex].
- This is also a linear function but with a slope of 2 and a y-intercept of 1.
- To graph this, plot the point where [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 2(1) + 1 = 3 \)[/tex].
- Select another point to ensure the line is correct. For example, choose [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2(2) + 1 = 5 \)[/tex].
So, the line for [tex]\( x \geq 1 \)[/tex] passes through the points (1, 3) and (2, 5).
2. Graph the Lines on the Same Coordinate Plane:
- For [tex]\( x < 1 \)[/tex], draw the line segment from (0, 4) to the left, passing through (-1, 5).
- For [tex]\( x \geq 1 \)[/tex], draw the line starting from (1, 3) to the right.
3. Check Continuity at [tex]\( x = 1 \)[/tex]:
To determine whether the function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex], check the left-hand limit (as [tex]\( x \)[/tex] approaches 1 from the left) and the right-hand limit (as [tex]\( x \)[/tex] approaches 1 from the right):
- Left-hand limit:
[tex]\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x + 4) = -1 + 4 = 3 \][/tex]
- Right-hand limit:
[tex]\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x + 1) = 2(1) + 1 = 3 \][/tex]
- Since both limits are the same and equal to [tex]\( f(1) = 3 \)[/tex], the function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex].
### Summary
The graph of the piecewise function:
- For [tex]\( x < 1 \)[/tex], the line segment is [tex]\( f(x) = -x + 4 \)[/tex].
- For [tex]\( x \geq 1 \)[/tex], the line segment is [tex]\( f(x) = 2x + 1 \)[/tex].
The function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex] because the limits from both sides are equal to the function's value at that point.
To visualize, you should draw both linear segments on the same graph, marking a dashed vertical line at [tex]\( x = 1 \)[/tex] if you’d like to emphasize the point of piecewise change.
The function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[ f(x) = \begin{cases} -x + 4 & \text{if } x < 1 \\ 2x + 1 & \text{if } x \geq 1 \end{cases} \][/tex]
### Step-by-Step Solution:
1. Graphing the Piecewise Functions:
a. For [tex]\( x < 1 \)[/tex]:
The function is [tex]\( f(x) = -x + 4 \)[/tex].
- This is a linear function with a slope of -1 and a y-intercept of 4.
- To graph this, plot the point where [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = -0 + 4 = 4 \)[/tex].
- Because it’s a line, select another point to better draw the line. For example, choose [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = -(-1) + 4 = 5 \)[/tex].
So, the line for [tex]\( x < 1 \)[/tex] passes through the points (0, 4) and (-1, 5).
b. For [tex]\( x \geq 1 \)[/tex]:
The function is [tex]\( f(x) = 2x + 1 \)[/tex].
- This is also a linear function but with a slope of 2 and a y-intercept of 1.
- To graph this, plot the point where [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 2(1) + 1 = 3 \)[/tex].
- Select another point to ensure the line is correct. For example, choose [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2(2) + 1 = 5 \)[/tex].
So, the line for [tex]\( x \geq 1 \)[/tex] passes through the points (1, 3) and (2, 5).
2. Graph the Lines on the Same Coordinate Plane:
- For [tex]\( x < 1 \)[/tex], draw the line segment from (0, 4) to the left, passing through (-1, 5).
- For [tex]\( x \geq 1 \)[/tex], draw the line starting from (1, 3) to the right.
3. Check Continuity at [tex]\( x = 1 \)[/tex]:
To determine whether the function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex], check the left-hand limit (as [tex]\( x \)[/tex] approaches 1 from the left) and the right-hand limit (as [tex]\( x \)[/tex] approaches 1 from the right):
- Left-hand limit:
[tex]\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x + 4) = -1 + 4 = 3 \][/tex]
- Right-hand limit:
[tex]\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x + 1) = 2(1) + 1 = 3 \][/tex]
- Since both limits are the same and equal to [tex]\( f(1) = 3 \)[/tex], the function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex].
### Summary
The graph of the piecewise function:
- For [tex]\( x < 1 \)[/tex], the line segment is [tex]\( f(x) = -x + 4 \)[/tex].
- For [tex]\( x \geq 1 \)[/tex], the line segment is [tex]\( f(x) = 2x + 1 \)[/tex].
The function [tex]\( f(x) \)[/tex] is continuous at [tex]\( x = 1 \)[/tex] because the limits from both sides are equal to the function's value at that point.
To visualize, you should draw both linear segments on the same graph, marking a dashed vertical line at [tex]\( x = 1 \)[/tex] if you’d like to emphasize the point of piecewise change.
