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The piecewise function [tex]\( f(x) \)[/tex] is defined as follows:

[tex]\[
f(x) = \begin{cases}
2x - 1 & \text{if } x \ \textless \ 0 \\
0 & \text{if } x = 0 \\
-2x + 1 & \text{if } x \ \textgreater \ 0
\end{cases}
\][/tex]

Which is the graph of [tex]\( f(x) \)[/tex]?



Answer :

GinnyAnswer
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.

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