To identify the graph that represents the given piecewise function
[tex]\[
f(x)=\begin{cases}
-2x & \text{if } x < -1 \\
-1 & \text{if } -1 \leq x < 2 \\
x-1 & \text{if } x \geq 2
\end{cases}
\][/tex]
we need to analyze each piece of the function and understand its behavior in the different intervals:
1. For [tex]\( x < -1 \)[/tex]: The function is defined as [tex]\( f(x) = -2x \)[/tex].
- This part of the function is a straight line with a negative slope of -2.
- For example:
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -2(-2) = 4 \)[/tex].
- When [tex]\( x = -1.5 \)[/tex], [tex]\( f(x) = -2(-1.5) = 3 \)[/tex].
- The line will pass through points like (-2, 4) and (-1, 2).
2. For [tex]\( -1 \leq x < 2 \)[/tex]: The function is constant and equal to [tex]\( -1 \)[/tex].
- This part of the function is a horizontal line.
- It remains [tex]\( f(x) = -1 \)[/tex] for all [tex]\(x\)[/tex] in the interval [tex]\([-1, 2)\)[/tex].
- This segment includes [tex]\( x = -1 \)[/tex] (closed interval) but not [tex]\( x = 2 \)[/tex] (open interval).
- The points will be continuous from (-1, -1) to just before (2, -1).
3. For [tex]\( x \geq 2 \)[/tex]: The function is defined as [tex]\( f(x) = x - 1 \)[/tex].
- This part of the function is a straight line with a slope of 1 and a y-intercept of -1.
- For example:
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 2 - 1 = 1 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 3 - 1 = 2 \)[/tex].
- The line starts at (2,1) and continues through points like (3, 2).
Based on this analysis, let's summarize the key points for each interval:
- [tex]\( x < -1 \)[/tex]: Line segment with equation [tex]\( f(x) = -2x \)[/tex]. Examples: (-2, 4), (-1.5, 3), (-1, 2).
- [tex]\( -1 \leq x < 2 \)[/tex]: Constant line [tex]\( f(x) = -1 \)[/tex]. Segment: from (-1, -1) to just before (2, -1).
- [tex]\( x \geq 2 \)[/tex]: Line segment with equation [tex]\( f(x) = x - 1 \)[/tex]. Examples: (2, 1), (3, 2).
Make sure any possible graph adheres closely to these descriptions, especially noting:
- The slope of lines (-2x, x-1) outside intervals.
- The constant value within [tex]\([-1, 2)\)[/tex].
- The boundaries' locations and continuity considerations.
The graph should distinctly connect pieces smoothly and indicate open or closed nature at points specified. This method ensures the piecmeal transition between defined behaviours uniformly.
