Overview
This lecture explains how to graph piecewise functions by breaking each function into its component parts and analyzing their domains separately.
Graphing Piecewise Functions
- A piecewise function has different expressions based on the value of x.
- Graph each part of the function only over its specified domain.
- Use open circles for excluded endpoints and closed circles for included endpoints on the graph.
Example 1: f(x) = x for x < 0, f(x) = 5 for x ≥ 0
- y = x is a straight line with slope 1, graphed for x < 0.
- y = 5 is a horizontal line, graphed for x ≥ 0.
- Use an open circle at (0, 0) and a closed circle at (0, 5).
Example 2: f(x) = 2 for x < 1, f(x) = x + 3 for x > 2
- y = 2 is a horizontal line for x < 1, open circle at (1, 2).
- y = x + 3 is a line with slope 1 and y-intercept 3, starts at (2, 5) with an open circle.
- No closed circles since inequalities do not include endpoints.
Example 3: f(x) = 2x + 1 for x < 1; f(x) = 1 for x = 1; f(x) = –x² for x > 1
- y = 2x + 1 is graphed left of x = 1, open circle at (1, 3).
- y = 1 is a single point at (1, 1), closed circle.
- y = –x² is a downward parabola for x > 1, starts at (1, –1), open circle.
Example 4: f(x) = 3x + 4 for x < 0; f(x) = 2 for x = 0; f(x) = √x for x > 1
- y = 3x + 4 is a line for x < 0, open circle at (0, 4).
- Single point at (0, 2), closed circle.
- y = √x is a curve for x > 1, open circle at (1, 1), passes through (4, 2).
Example 5: f(x) = 1/x for x < 0; f(x) = 3 for 0 ≤ x < 3; f(x) = –x + 5 for x ≥ 3
- y = 1/x for x < 0, only the left side of the hyperbola.
- y = 3 is a horizontal line for 0 ≤ x < 3, closed at (0, 3), open at (3, 3).
- y = –x + 5 for x ≥ 3, line with slope –1 and y-intercept 5, closed circle at (3, 2).
Key Terms & Definitions
- Piecewise Function — A function defined by different expressions over different intervals of x.
- Open Circle — Indicates a point not included in the graph (inequality does not include endpoint).
- Closed Circle — Indicates a point is included in the graph (inequality includes endpoint).
- Slope (m) — Change in y over change in x in a linear equation y = mx + b.
- Y-intercept (b) — The y-value where the graph crosses the y-axis.
Action Items / Next Steps
- Practice graphing piecewise functions using different expressions and domains.
- Review how to determine open versus closed circles for each interval.
- Complete assigned exercises on graphing piecewise functions.