Answer :
Let's analyze the function [tex]\( f(x) \)[/tex] which is defined in a piecewise manner:
[tex]\[ f(x)=\left\{ \begin{array}{ll} -x, & x < 0 \\ 1, & x \geq 0 \end{array} \right. \][/tex]
### Step-by-Step Solution
1. Understanding the Piecewise Function:
- For [tex]\( x < 0 \)[/tex], the function takes the form [tex]\( f(x) = -x \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], the function is constant, [tex]\( f(x) = 1 \)[/tex].
2. Analyzing the Transition Point:
- The point of transition between the pieces of the function is at [tex]\( x = 0 \)[/tex].
- To correctly graph the function, we need to examine the behavior at [tex]\( x = 0 \)[/tex].
3. Evaluating Limits Around [tex]\( x = 0 \)[/tex]:
- Left-Hand Limit at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0^-) \][/tex]
When [tex]\( x \)[/tex] approaches 0 from the negative side ([tex]\( x < 0 \)[/tex]):
[tex]\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x = -0 = 0 \][/tex]
- Right-Hand Limit at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0^+) \][/tex]
When [tex]\( x \)[/tex] approaches 0 from the positive side ([tex]\( x \geq 0 \)[/tex]):
[tex]\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 1 = 1 \][/tex]
4. Identifying the Nature of the Transition:
- At [tex]\( x = 0 \)[/tex], from the left-hand side [tex]\( f(x) \)[/tex] approaches 0, whereas from the right-hand side it jumps to 1.
- This means the function [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = 0 \)[/tex].
5. Determining Where to Draw the Open Circle:
- An open circle is used to indicate a point where the function value changes abruptly.
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0^-) = 0 \)[/tex], indicating that the function was approaching 0 right up until [tex]\( x = 0 \)[/tex] from the left-hand side.
- Thus, we should draw an open circle at the point where [tex]\( x=0 \)[/tex] and the left-hand limit holds, which is [tex]\((0, 0)\)[/tex].
### Conclusion
After carefully analyzing the piecewise function and the transition at [tex]\( x = 0 \)[/tex], we determine that the open circle should be drawn at:
[tex]\[(0, 0)\][/tex]
So, the correct choice is:
[tex]\[(0, 0)\][/tex]
[tex]\[ f(x)=\left\{ \begin{array}{ll} -x, & x < 0 \\ 1, & x \geq 0 \end{array} \right. \][/tex]
### Step-by-Step Solution
1. Understanding the Piecewise Function:
- For [tex]\( x < 0 \)[/tex], the function takes the form [tex]\( f(x) = -x \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], the function is constant, [tex]\( f(x) = 1 \)[/tex].
2. Analyzing the Transition Point:
- The point of transition between the pieces of the function is at [tex]\( x = 0 \)[/tex].
- To correctly graph the function, we need to examine the behavior at [tex]\( x = 0 \)[/tex].
3. Evaluating Limits Around [tex]\( x = 0 \)[/tex]:
- Left-Hand Limit at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0^-) \][/tex]
When [tex]\( x \)[/tex] approaches 0 from the negative side ([tex]\( x < 0 \)[/tex]):
[tex]\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x = -0 = 0 \][/tex]
- Right-Hand Limit at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0^+) \][/tex]
When [tex]\( x \)[/tex] approaches 0 from the positive side ([tex]\( x \geq 0 \)[/tex]):
[tex]\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 1 = 1 \][/tex]
4. Identifying the Nature of the Transition:
- At [tex]\( x = 0 \)[/tex], from the left-hand side [tex]\( f(x) \)[/tex] approaches 0, whereas from the right-hand side it jumps to 1.
- This means the function [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = 0 \)[/tex].
5. Determining Where to Draw the Open Circle:
- An open circle is used to indicate a point where the function value changes abruptly.
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0^-) = 0 \)[/tex], indicating that the function was approaching 0 right up until [tex]\( x = 0 \)[/tex] from the left-hand side.
- Thus, we should draw an open circle at the point where [tex]\( x=0 \)[/tex] and the left-hand limit holds, which is [tex]\((0, 0)\)[/tex].
### Conclusion
After carefully analyzing the piecewise function and the transition at [tex]\( x = 0 \)[/tex], we determine that the open circle should be drawn at:
[tex]\[(0, 0)\][/tex]
So, the correct choice is:
[tex]\[(0, 0)\][/tex]
