To determine where an open circle should be drawn for the given piecewise function
[tex]\[ f(x) = \begin{cases}
-x & \text{if } x < 0 \\
1 & \text{if } x \geq 0
\end{cases}
\][/tex]
we need to examine the point of transition, which occurs at [tex]\(x = 0\)[/tex].
1. Evaluate the function at [tex]\( x = 0 \)[/tex]:
- For [tex]\( x < 0 \)[/tex], the function is defined as [tex]\( f(x) = -x \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], the function is defined as [tex]\( f(x) = 1 \)[/tex].
2. At the transition point [tex]\( x = 0 \)[/tex]:
- When approaching [tex]\( x = 0 \)[/tex] from the left (i.e., [tex]\( x \to 0^- \)[/tex]), we use the definition [tex]\( f(x) = -x \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] yields [tex]\( f(0^-) = -0 = 0 \)[/tex]. This tells us that just before [tex]\( x = 0 \)[/tex], the function approaches the value [tex]\(0\)[/tex].
- When at [tex]\( x = 0 \)[/tex], according to the piecewise definition, we should use [tex]\( f(x) = 1 \)[/tex] for [tex]\( x \geq 0 \)[/tex].
- Thus, [tex]\( f(0) = 1 \)[/tex].
3. Open Circle and Closed Circle:
- Since the function [tex]\( f(x) = -x \)[/tex] is not defined at [tex]\( x = 0 \)[/tex] (it's defined for [tex]\( x < 0 \)[/tex]), we need to place an open circle there to indicate that the value approaches 0 but does not include it.
- Thus, the open circle at the point of transition represents the value of [tex]\( f(x) \)[/tex] right before [tex]\( x = 0 \)[/tex].
Given these observations, the open circle should be drawn at the point where the left-hand limit (approach from [tex]\( x < 0 \)[/tex]) meets the transition, which is [tex]\( (0, 0) \)[/tex].
Therefore, at what point should the open circle be drawn?
The open circle should be drawn at the point [tex]\((0, 0)\)[/tex].
