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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.
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