Boundary Points in Piecewise Functions
Boundary points in piecewise functions are graphed with closed circles (●) for included domain values and open circles (○) for excluded values — an essential graphing detail in enVision Algebra 1 Chapter 5 for Grade 11. For f(x) = {x+1 if x<2, 3x-2 if x≥2}, the piece x+1 gives (2,3) with an open circle (x=2 is excluded), while 3x-2 gives (2,4) with a closed circle (x=2 is included). For g(x) = {x² if x≤-1, 2x if x>-1}, the first piece gets a closed circle at (-1,1) and the second an open circle at (-1,-2). Correct circle notation prevents ambiguity about which piece applies at the boundary.
Key Concepts
When graphing piecewise functions, use closed circles ($\bullet$) to indicate points included in the domain and open circles ($\circ$) to indicate points excluded from the domain at boundary values.
Common Questions
What determines whether a boundary point gets an open or closed circle?
Check the inequality for that piece. If the inequality is ≤ or ≥ (includes the boundary), use a closed circle. If it is < or > (excludes the boundary), use an open circle.
For f(x) = {x+1 if x<2, 3x-2 if x≥2}, what circles appear at x=2?
Open circle at (2, 3) for the first piece (x<2 excludes 2), and closed circle at (2, 4) for the second piece (x≥2 includes 2). The function value at x=2 is 4.
Can a piecewise function have both a closed and open circle at the same x-value?
Yes. Each piece may give a different y-value at the boundary x. One piece can be included (closed circle) and the other excluded (open circle), showing the function is defined at that boundary by one specific piece.
What happens if both pieces have the same y-value at the boundary?
The function is continuous at that boundary. Both circles would coincide at the same point, so typically just one closed circle is shown, indicating a smooth transition.
How do boundary circles relate to whether a piecewise function is continuous?
If an open circle and closed circle at the same boundary x sit at different y-values, the function has a jump discontinuity there. If they meet at the same y-value, the function is continuous.