Piecewise functions are used in real-world applications:
Cell phone billing
Shipping rates (e.g., UPS, USPS)
Example: Sprint Cell Phone Plan (2005)
Monthly bill modeled by a graph
x-axis: Number of minutes used per month
y-axis: Monthly bill in dollars
Calculating the Bill
360 minutes: Locate 360 on x-axis, go to graph, y = $40
700 minutes: Closed point at y = $55, open point not defined
Analyzing the Bill
If monthly bill = $55, customer usage:
Horizontal interval where bill = $55
640 minutes results in a $55 bill
600 minutes leads to a $50 bill
600.001 minutes jumps to $55
Expressing the interval:
Let t = number of minutes
t > 600 and t <= 700
Open interval at 600, closed at 700
Graphing Piecewise Functions
Example
Function: g(x)
g(x) = 2 for x >= 1 (horizontal line at y = 2)
g(x) = -x - 2 for x < 1 (line with slope -1, y-intercept -2)
Combine graphs onto the same plane
Another Example
Function: f(x) with three pieces
Piece 1: f(x) = 3 when x <= 0 (horizontal line at y = 3)
Piece 2: f(x) = 1 - x^2 from 0 < x <= 2 (parabola segment)
Piece 3: f(x) = 2x - 4 for x > 2 (line, open circle at x = 2)
Graphing on Graphing Calculators
Procedure
Enter each function rule for each piece separately
Multiply by the interval condition in parentheses
Example:
First piece: 3 * (x <= 0)
Second piece: (1 - x^2) * (0 < x <= 2)
Third piece: (2x - 4) * (x > 2)*
Inequality Symbols
Access via second math (test menu)
Graphing calculators do not display open/closed points
Users must determine open/closed points from graphs
The lecture covered interpreting and graphing piecewise functions, their real-life applications, example scenarios, and methods to graph them using a calculator.