Overview
This lecture covers piecewise functions, how to evaluate and graph them, and how to determine their domains and ranges.
Piecewise Functions: Introduction
- A piecewise function is defined by different formulas on different intervals of its domain.
- Each input fits exactly one part of the function, resulting in a single output.
Evaluating Piecewise Functions
- To find a value like f(3), identify which interval 3 falls into and use the corresponding formula.
- Example: For (f(t)), if ( -5 \leq 3 \leq 5 ), use (f(3) = 3^2 = 9).
Graphing Piecewise Functions
- Graph each piece separately according to its interval.
- The graph may have "splits" or breaks at the interval boundaries.
- Use open or closed circles to indicate whether endpoints are included.
Example 1: Three-Part Piecewise Function
- Given (f(t) = 3) for ( t < -5 ), (t^2) for ( -5 \leq t \leq 5 ), (t+1) for ( t > 5 ).
- Graph is: horizontal line for ( t < -5 ), parabola for ( -5 \leq t \leq 5 ), line with slope 1 for ( t > 5 ).
- The domain is all real numbers ((-\infty, \infty)).
- The range is ([0, \infty)) because values start at 0 and go up without bound.
Example 2: Two-Part Piecewise Function
- Given (f(x) = x-1) for ( x \leq 0 ), (x+1) for ( x > 0 ).
- At ( x=0 ), (f(0) = -1); left side is filled, right side has an open circle at (0,1).
- The graph consists of two lines with a gap between outputs -1 and 1.
- The domain is all real numbers.
- The range is ((-\infty, -1)] ∪ ((1, \infty)) — note the gap between -1 and 1.
Key Terms & Definitions
- Piecewise Function — A function defined by different expressions for different intervals of its domain.
- Open Circle — Indicates an endpoint not included in the graph.
- Closed Circle — Indicates an endpoint included in the graph.
- Domain — Set of all possible input values.
- Range — Set of all possible output values.
Action Items / Next Steps
- Practice evaluating and graphing piecewise functions.
- Be ready to identify domains and ranges for piecewise graphs.
- Ask questions if concepts remain unclear.