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: $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 3 & \text{if } x \geq 0 \end{cases}$

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 $x = -1$ and $x = 2$ : $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 3 & \text{if } x \geq 0 \end{cases}$

Step-by-Step Solution:

For $x = -1$ : Since $-1 < 0$ , use the first part of the function: $f(-1) = (-1)^2 = 1$ .
For $x = 2$ : Since $2 \geq 0$ , use the second part of the function: $f(2) = 2(2) + 3 = 7$ .

Validation: For $x = -1$ , original: $1$ ; Simplified: $1$ ✓
For $x = 2$ , original: $7$ ; Simplified: $7$ ✓

Example 2 (Intermediate)

Problem: Evaluate the piecewise function at $x = -2$ and $x = 1$ : $g(x) = \begin{cases} -x + 5 & \text{if } x \leq 0 \\ x^2 - 2 & \text{if } x > 0 \end{cases}$

Step-by-Step Solution:

For $x = -2$ : Since $-2 \leq 0$ , use the first part of the function: $g(-2) = -(-2) + 5 = 7$ .
For $x = 1$ : Since $1 > 0$ , use the second part of the function: $g(1) = 1^2 - 2 = -1$ .

Validation: For $x = -2$ , original: $7$ ; Simplified: $7$ ✓
For $x = 1$ , original: $-1$ ; Simplified: $-1$ ✓

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.