Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To graph the piecewise function [tex]\( f(x) \)[/tex], we need to graph three different expressions depending on the value of [tex]\( x \)[/tex]. Let's break down each piece and then combine them into the overall graph.
1. For [tex]\( x < 0 \)[/tex]: The expression is [tex]\( f(x) = 2x - 1 \)[/tex].
This is a linear function with a slope of 2 and a y-intercept of -1. We can plot some points to see the shape of this line:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2(-1) - 1 = -3 \)[/tex].
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 2(-2) - 1 = -5 \)[/tex].
- When [tex]\( x = -0.5 \)[/tex], [tex]\( f(x) = 2(-0.5) - 1 = -2 \)[/tex].
These points, along with the general behavior of a line with slope 2, will help us sketch this part of the function.
2. For [tex]\( x = 0 \)[/tex]: The expression is [tex]\( f(x) = 0 \)[/tex].
At [tex]\( x = 0 \)[/tex], the value of the function is simply 0. This gives us the point (0, 0) on the graph.
3. For [tex]\( x > 0 \)[/tex]: The expression is [tex]\( f(x) = -2x + 1 \)[/tex].
This is another linear function, but with a slope of -2 and a y-intercept of 1. We can plot some points for this line:
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -2(1) + 1 = -1 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -2(2) + 1 = -3 \)[/tex].
- When [tex]\( x = 0.5 \)[/tex], [tex]\( f(x) = -2(0.5) + 1 = 0 \)[/tex].
These points show the behavior of the line with slope -2 for [tex]\( x > 0 \)[/tex].
Combining all these pieces:
- For values of [tex]\( x \)[/tex] less than 0, plot the line [tex]\( f(x) = 2x - 1 \)[/tex].
- At [tex]\( x = 0 \)[/tex], plot the point (0, 0).
- For values of [tex]\( x \)[/tex] greater than 0, plot the line [tex]\( f(x) = -2x + 1 \)[/tex].
Now, let’s put this all together in a graph:
1. Plot [tex]\( f(x) = 2x - 1 \)[/tex] for [tex]\( x < 0 \)[/tex].
```
Points: (-1, -3), (-2, -5), (-0.5, -2), ...
```
2. Plot [tex]\( f(x) = 0 \)[/tex] at [tex]\( x = 0 \)[/tex].
```
Point: (0, 0)
```
3. Plot [tex]\( f(x) = -2x + 1 \)[/tex] for [tex]\( x > 0 \)[/tex].
```
Points: (1, -1), (2, -3), (0.5, 0), ...
```
With these points and the descriptions combined, the final graph should look like this:
- Left of [tex]\( x = 0 \)[/tex]: The line [tex]\( y = 2x - 1 \)[/tex] decreases and passes through points like (-1, -3).
- At [tex]\( x = 0 \)[/tex]: There is a single point (0, 0).
- Right of [tex]\( x = 0 \)[/tex]: The line [tex]\( y = -2x + 1 \)[/tex] decreases and passes through points like (1, -1).
Here is a sketch of the final graph:
```
y
|
|
|
|
|
|
|----------------- x
|
|
|
|
|
|
```
Thus, we have plotted the piecewise function [tex]\( f(x) \)[/tex] correctly according to its definition.
1. For [tex]\( x < 0 \)[/tex]: The expression is [tex]\( f(x) = 2x - 1 \)[/tex].
This is a linear function with a slope of 2 and a y-intercept of -1. We can plot some points to see the shape of this line:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2(-1) - 1 = -3 \)[/tex].
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 2(-2) - 1 = -5 \)[/tex].
- When [tex]\( x = -0.5 \)[/tex], [tex]\( f(x) = 2(-0.5) - 1 = -2 \)[/tex].
These points, along with the general behavior of a line with slope 2, will help us sketch this part of the function.
2. For [tex]\( x = 0 \)[/tex]: The expression is [tex]\( f(x) = 0 \)[/tex].
At [tex]\( x = 0 \)[/tex], the value of the function is simply 0. This gives us the point (0, 0) on the graph.
3. For [tex]\( x > 0 \)[/tex]: The expression is [tex]\( f(x) = -2x + 1 \)[/tex].
This is another linear function, but with a slope of -2 and a y-intercept of 1. We can plot some points for this line:
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -2(1) + 1 = -1 \)[/tex].
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -2(2) + 1 = -3 \)[/tex].
- When [tex]\( x = 0.5 \)[/tex], [tex]\( f(x) = -2(0.5) + 1 = 0 \)[/tex].
These points show the behavior of the line with slope -2 for [tex]\( x > 0 \)[/tex].
Combining all these pieces:
- For values of [tex]\( x \)[/tex] less than 0, plot the line [tex]\( f(x) = 2x - 1 \)[/tex].
- At [tex]\( x = 0 \)[/tex], plot the point (0, 0).
- For values of [tex]\( x \)[/tex] greater than 0, plot the line [tex]\( f(x) = -2x + 1 \)[/tex].
Now, let’s put this all together in a graph:
1. Plot [tex]\( f(x) = 2x - 1 \)[/tex] for [tex]\( x < 0 \)[/tex].
```
Points: (-1, -3), (-2, -5), (-0.5, -2), ...
```
2. Plot [tex]\( f(x) = 0 \)[/tex] at [tex]\( x = 0 \)[/tex].
```
Point: (0, 0)
```
3. Plot [tex]\( f(x) = -2x + 1 \)[/tex] for [tex]\( x > 0 \)[/tex].
```
Points: (1, -1), (2, -3), (0.5, 0), ...
```
With these points and the descriptions combined, the final graph should look like this:
- Left of [tex]\( x = 0 \)[/tex]: The line [tex]\( y = 2x - 1 \)[/tex] decreases and passes through points like (-1, -3).
- At [tex]\( x = 0 \)[/tex]: There is a single point (0, 0).
- Right of [tex]\( x = 0 \)[/tex]: The line [tex]\( y = -2x + 1 \)[/tex] decreases and passes through points like (1, -1).
Here is a sketch of the final graph:
```
y
|
|
|
|
|
|
|----------------- x
|
|
|
|
|
|
```
Thus, we have plotted the piecewise function [tex]\( f(x) \)[/tex] correctly according to its definition.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
