Page 2: Evaluation Techniques

This page demonstrates piecewise functions evaluation examples through practical problems. It shows how to determine which piece of the function to use based on the input value.

Highlight: When evaluating piecewise functions, first identify which interval contains your input value, then use the corresponding function piece.

Example: For f$-5$, since -5 < -3, use the first piece: 2x-5 Solution: 2$-5$-5 = -10-5 = -15

Page 3: Graphical Analysis

This page presents a graphical representation of a piecewise function, emphasizing the visual interpretation of function behavior.

Highlight: The graph shows distinct pieces connecting at transition points, illustrating the continuous or discontinuous nature of the function.

Example: Finding f(0) = 2 demonstrates how to read function values directly from the graph.

Page 4: Domain Restrictions

This page explores piecewise functions with specific domain restrictions and their graphical representations.

Vocabulary: Domain restrictions define the intervals where each piece of the function is valid.

Example: The function transitions at x = -1 and x = 3, creating distinct regions of behavior.

Page 5: Linear Piecewise Functions

This page focuses on piecewise functions composed of linear pieces with different slopes and y-intercepts.

Example: The function consists of three pieces:

• f$x$ = 2x+5 for x ≤ 0
• f$x$ = -2x+12 for 0 < x ≤ 4
• A constant value for x > 4

Page 6: Step Functions

This page introduces step functions as a special case of piecewise functions, showing discontinuous behavior at transition points.

Definition: A step function is a piecewise function that has constant values over specific intervals.

Page 7: Point Functions

This page demonstrates piecewise functions defined at specific points, including isolated points and intervals.

Highlight: Some piecewise functions may include individual points with distinct values, creating discontinuities.

Page 8: Quadratic Pieces

This page explores piecewise functions containing quadratic expressions, showing how parabolic sections can be combined with other function types.

Example: The function includes a quadratic piece defined over the interval -2 < x < 2, demonstrating how curved sections can be incorporated into piecewise functions.

Page 1: Introduction to Piecewise Functions

This page introduces the fundamental concept of piecewise functions. A piecewise function consists of multiple sub-functions, each defined over specific intervals of the domain.

Definition: A piecewise function is a mathematical function composed of multiple sub-functions, where each piece applies to a specific interval of the input values.

Example: The function presented shows three distinct pieces:

• f$x$ = 2x-5 for x < -3
• f$x$ = -x² + 4x for -3 < x < 7
• f$x$ = √2x+6 for x ≥ 7