Lecture Notes: Vertical Line Test and Piecewise Functions
Overview
This lecture covers two main topics:
- Understanding the Vertical Line Test
- Drawing and analyzing a piecewise function with three parts.
1. Vertical Line Test
Definition:
- The vertical line test is a method to determine if a graph represents a function.
- A graph passes the vertical line test if no vertical line intersects the graph at more than one point.
Explanation:
- Example: The equation (y^2 = x + 1) implies that for (x = 0), (y = \pm 1), indicating two values of y for a single x.
- Graphically, for a curve like (x = y^2 - 1), a vertical line at (x = 0) intersects the graph at two points ((0, 1)) and ((0, -1)).
- If a vertical line crosses a graph at multiple points, the graph is not a function.
Significance:
- Graphs that pass the vertical line test (e.g., (y = x^3, y = \sqrt{x})) are functions because they can be expressed as (y) as a function of (x).
- If any vertical line intersects more than once, the curve fails the vertical line test and is not a function.
2. Piecewise Functions
Definition:
- A piecewise function is defined by different expressions for different intervals of the domain.
Example Function:
- Define a function as:
- (-2x) for (x \leq 1)
- (x^3) for (1 < x < 2)
- Constant 1 for (x = 2)
Graphing Process:
-
Segment 1:
- For (x \leq 1), graph (-2x).
- Points: ((1, -2)) (closed circle), ((0, 0))
-
Segment 2:
- For (1 < x < 2), graph (x^3).
- Open circles at ((1, 1)) and ((2, 8)).
- Plot intermediate points and note the open nature indicating exclusion.
-
Segment 3:
- For (x = 2), graph is a constant 1.
- Solid circle at ((2, 1)).
Characteristics:
- The graph does not fail the vertical line test as no vertical line intersects more than once.
- Open circles (non-inclusive) and closed circles (inclusive) are used to denote specific points along the piecewise function.
Conclusion
- The lecture concludes with preparation to move to applications of lines and linear functions in subsequent material.
This lecture provides a foundational understanding of the vertical line test's importance in determining functions and introduces piecewise functions, their construction, and graphical representation.