Piecewise Functions and Their Evaluation

Piecewise Functions

Definition and Structure

Piecewise functions are mathematical functions defined by different expressions depending on the input value. They are commonly used to model situations where a rule or relationship changes based on the domain of the input variable.

• Piecewise Function: A function composed of multiple sub-functions, each applied to a certain interval of the domain.

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

General Form:

$f(x) = \begin{cases} f_1(x), & \text{if } x \in D_1 \\ f_2(x), & \text{if } x \in D_2 \\ \vdots \\ f_n(x), & \text{if } x \in D_n \end{cases}$

Evaluating Piecewise Functions

To evaluate a piecewise function at a given value, determine which interval the input belongs to and use the corresponding expression.

• Step 1: Identify the interval that contains the input value.

• Step 2: Substitute the input value into the corresponding sub-function.

• Step 3: Compute the result.

Example:

Suppose we have the following piecewise function:

$f(x) = \begin{cases} x^2, & \text{if } x < 0 \\ 2x + 1, & \text{if } 0 \leq x < 3 \\ 5, & \text{if } x \geq 3 \end{cases}$

• Evaluate $f(-2)$: Since $-2 < 0$, use $x^2$: $f(-2) = (-2)^2 = 4$.

• Evaluate $f(1)$: Since $0 \leq 1 < 3$, use $2x + 1$: $f(1) = 2(1) + 1 = 3$.

• Evaluate $f(4)$: Since $4 \geq 3$, use $5$:$f(4) = 5$.

Applications of Piecewise Functions

Piecewise functions are used in various real-world contexts, such as:

• Tax brackets: Different tax rates apply to different income ranges.

• Shipping costs: Cost may change based on weight intervals.

• Physics: Modeling motion with different forces in different regions.

Properties and Graphs

Graphs of piecewise functions often have breaks, jumps, or changes in slope at the boundaries between intervals.

• Continuity: A piecewise function may or may not be continuous at the boundaries.

• Domain: The domain is the union of all intervals specified in the definition.

• Range: The range depends on the outputs of all sub-functions.

Table: Example Piecewise Function Evaluation

x	Interval	Expression Used	f(x)
-2	$x < 0$	$x^2$	4
1	$0 \leq x < 3$	$2x + 1$	3
4	$x \geq 3$	5	5

Summary

• Piecewise functions allow for different rules over different intervals.

• Evaluating requires careful attention to the input value's interval.

• Applications are common in real-world scenarios where rules change based on conditions.

Additional info: The original file was highly fragmented and partially illegible, but the recurring mention of 'piecewise function' and 'evaluate at given values' indicates a focus on evaluating piecewise functions, a standard Precalculus topic. The table and examples were inferred for completeness.