Evaluate Piecewise Functions

Algebra-1

1. Fundamental Concepts

Definition: Piecewise functions are defined by different expressions over different intervals of the domain.
Domain: The set of all possible input values (x-values) for which the function is defined.
Range: The set of all possible output values (y-values) that the function can produce.

2. Key Concepts

Evaluation Rule:

Graphical Interpretation: The graph of a piecewise function consists of segments defined by each expression within its respective interval.

Application: Used in modeling situations where different rules apply under different conditions, such as tax brackets or pricing structures.

3. Examples

Example 1 (Basic)

Problem: Evaluate the piecewise function at and :

Step-by-Step Solution:

For : Since , use the first part of the function: .
For : Since , use the second part of the function: .

Validation: For , original: ; Simplified: ✓
For , original: ; Simplified: ✓

Example 2 (Intermediate)

Problem: Evaluate the piecewise function at and :

Step-by-Step Solution:

For : Since , use the first part of the function: .
For : Since , use the second part of the function: .

Validation: For , original: ; Simplified: ✓
For , original: ; Simplified: ✓

4. Problem-Solving Techniques

Identify Intervals: Determine the intervals for which each piece of the function is defined.
Substitute Values: Substitute the given value into the appropriate piece of the function.
Check Boundaries: Pay special attention to the boundaries between intervals to ensure correct evaluation.