To graph the piecewise function [tex]\( f(x) \)[/tex], let's examine each part of the piecewise function separately:
1. For [tex]\( x < 0 \)[/tex]:
The function is given by [tex]\( f(x) = 2x - 1 \)[/tex].
- This is a linear function with a slope of 2 and a y-intercept of [tex]\(-1\)[/tex].
- When [tex]\( x \)[/tex] is slightly less than 0, [tex]\( f(x) \)[/tex] is slightly less than [tex]\(-1\)[/tex].
- As [tex]\( x \)[/tex] becomes more negative, [tex]\( f(x) \)[/tex] decreases more steeply due to the positive slope.
- This part of the function is defined for all negative values of [tex]\( x \)[/tex].
2. For [tex]\( x = 0 \)[/tex]:
The function is defined as [tex]\( f(x) = 0 \)[/tex].
- At [tex]\( x = 0 \)[/tex], the value of the function is [tex]\( f(0) = 0 \)[/tex].
- This is a single point on the graph at the origin [tex]\((0, 0)\)[/tex].
3. For [tex]\( x > 0 \)[/tex]:
The function is given by [tex]\( f(x) = -2x + 1 \)[/tex].
- This is a linear function with a slope of [tex]\(-2\)[/tex] and a y-intercept of 1.
- When [tex]\( x \)[/tex] is slightly more than 0, [tex]\( f(x) \)[/tex] is slightly less than 1.
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases due to the negative slope.
- This part of the function is defined for all positive values of [tex]\( x \)[/tex].
To construct the graph:
- Start with the line [tex]\( f(x) = 2x - 1 \)[/tex] for [tex]\( x < 0 \)[/tex]. This line intersects the y-axis at [tex]\((0, -1)\)[/tex] and has a slope of 2, indicating it rises steeply to the right as [tex]\( x \)[/tex] increases toward 0 from the negative side.
- Plot the single point [tex]\((0, 0)\)[/tex] for [tex]\( x = 0 \)[/tex].
- Then, continue with the line [tex]\( f(x) = -2x + 1 \)[/tex] for [tex]\( x > 0 \)[/tex]. This line intersects the y-axis at [tex]\((0, 1)\)[/tex] and has a slope of [tex]\(-2\)[/tex], indicating it falls steeply to the right as [tex]\( x \)[/tex] increases from 0.
The overall graph will consist of:
- A linear segment with a positive slope to the left of the y-axis.
- A single point at the origin [tex]\((0, 0)\)[/tex].
- A linear segment with a negative slope to the right of the y-axis.
Keep in mind that these are three distinct parts: two separate lines and a point at the origin. The transitions at [tex]\( x = 0 \)[/tex] are smooth in terms of boundaries but not in terms of value:
- The left segment starts at [tex]\((-\infty, \infty)\)[/tex] and approaches the point [tex]\( (0, -1) \)[/tex].
- The right segment starts at the point [tex]\( (0, 1) \)[/tex] and proceeds to [tex]\((\infty, -\infty)\)[/tex].
However, at [tex]\( x = 0 \)[/tex], the function value is exactly 0, which is explicitly stated, differentiating it from nearby values on either side.
This complete step-by-step analysis encapsulates the feature of the piecewise function [tex]\( f(x) \)[/tex]. The graph 'jumps' at the point [tex]\( x = 0 \)[/tex], creating a discontinuity between [tex]\( -1 \)[/tex] and [tex]\(1\)[/tex] within very close vicinity, strictly only reaching the value 0 right at [tex]\( x = 0 \)[/tex].
