Piecewise Functions

Introduction to Piecewise Functions

Piecewise functions are an important concept in calculus prerequisites, as they allow us to define a function using different expressions depending on the input value. This is especially useful for modeling situations where a rule changes at certain points, and is foundational for understanding limits and continuity in calculus.

• Definition: A piecewise function is a single function composed of two or more equations, each valid for a specific interval of the domain.

• Notation: Piecewise functions are typically written using curly braces to indicate the different cases.

• Continuity: If the y-values (outputs) at the boundaries between pieces do not match, the function has a jump discontinuity at those points.

• Graphing: When graphing, use open or closed circles to indicate whether endpoints are included or excluded.

Example:

• Consider the function:

• To evaluate , , and , substitute each value into the appropriate equation:

  • (since )

  • (since )

  • (since )

Evaluating Piecewise Functions

To evaluate a piecewise function at a given input, determine which interval the input falls into and use the corresponding expression.

• Step 1: Identify the interval for the input value.

• Step 2: Substitute the input into the correct formula.

• Step 3: Compute the result.

Practice Example:

• Given:

• Evaluate :

  • Since , use .

• Evaluate :

  • Since , use .

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each segment according to its formula and domain. Pay attention to endpoints, using open circles for excluded values and closed circles for included values.

• Example:

• For , plot a horizontal line at .

• For , plot the line from to .

• For , plot the line for $ x \geq 1 $.

• At and , check which expressions include the endpoint and use closed or open circles accordingly.

Summary Table: Piecewise Function Structure

Applications: Piecewise functions are used to model tax brackets, shipping costs, and physical phenomena with abrupt changes.

Additional info: Piecewise functions are foundational for understanding limits, continuity, and later topics in calculus such as integration and differentiation of non-continuous functions.