The function [tex]$f(x)$[/tex] is to be graphed on a coordinate plane.
[tex]$f(x)=\left\{\begin{array}{ll}
-x, & x\ \textless \ 0 \\
1, & x \geq 0
\end{array}\right.$[/tex]
At what point should an open circle be drawn?
A. (-1,0)
B. (0,0)
C. (0,1)
D. (1,0)
Answer (1)
The function approaches (0,0) as x approaches 0 from the left.
The function is defined as f(0) = 1 at x=0, so the point (0,1) is included.
An open circle is used at (0,0) to show it's not included, while (0,1) is a closed point.
The open circle should be drawn at ( 0 , 0 ) .
Explanation
Understanding the Piecewise Function We are given a piecewise function: f ( x ) = { − x , x < 0 1 , x ≥ 0 We need to find the point where an open circle should be drawn on the graph of this function. An open circle is used to indicate a point that is not included in the graph. In this case, it occurs at x = 0 because the function value changes abruptly.
Analyzing the Behavior Around x=0 For x < 0 , the function is f ( x ) = − x . As x approaches 0 from the left, f ( x ) approaches − 0 = 0 . So, the function approaches the point ( 0 , 0 ) from the left. For x ≥ 0 , the function is f ( x ) = 1 . At x = 0 , f ( 0 ) = 1 . So, the function is defined at the point ( 0 , 1 ) .
Determining the Location of the Open Circle Since the function approaches ( 0 , 0 ) from the left but is defined as ( 0 , 1 ) at x = 0 , we use an open circle at ( 0 , 0 ) to indicate that this point is not included in the graph, and a closed circle (or a point) at ( 0 , 1 ) to indicate that this point is included in the graph.
Final Answer Therefore, the open circle should be drawn at the point ( 0 , 0 ) .
Examples
Imagine you are designing a simple on/off switch that behaves differently based on whether the input is positive or negative. This piecewise function is similar: for negative inputs, the output decreases linearly, while for positive inputs, the output is constant. Understanding where to place the 'open circle' helps you accurately represent the switch's behavior at the transition point, ensuring that the switch functions as intended without ambiguity. This concept is crucial in various engineering applications, such as control systems and signal processing, where precise behavior at transition points is essential.
