Learn on PengiBig Ideas Math, Algebra 1Chapter 4: Writing Linear Functions

Lesson 7: Piecewise Functions

Property.

Section 1

Definition and Notation of Piecewise Functions

Property

A piecewise function is defined by different equations over different intervals of its domain. The general notation is:

f(x)={equation1if condition1equation2if condition2⋮⋮equationnif conditionnf(x) = \begin{cases} \text{equation}_1 & \text{if condition}_1 \\ \text{equation}_2 & \text{if condition}_2 \\ \vdots & \vdots \\ \text{equation}_n & \text{if condition}_n \end{cases}

Examples

Section 2

Determining Domain Conditions for Piecewise Functions

Property

For a piecewise function f(x)={f1(x)if x satisfies condition 1f2(x)if x satisfies condition 2⋮⋮f(x) = \begin{cases} f_1(x) & \text{if } x \text{ satisfies condition 1} \\ f_2(x) & \text{if } x \text{ satisfies condition 2} \\ \vdots & \vdots \end{cases}, determine which piece applies by checking which condition the input value satisfies.

Examples

Section 3

Evaluating Piecewise Functions

Property

To evaluate a piecewise function at a given input value, first determine which piece (or condition) the input satisfies, then substitute the input into the corresponding function rule for that piece.

Examples

Section 4

Graphing Piecewise Functions

Property

To graph a piecewise function, we plot each piece separately on its specified domain, then combine them into one graph. Each piece is graphed only over its defined interval, and we use open or closed circles to indicate whether endpoints are included.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Definition and Notation of Piecewise Functions

Property

A piecewise function is defined by different equations over different intervals of its domain. The general notation is:

f(x)={equation1if condition1equation2if condition2⋮⋮equationnif conditionnf(x) = \begin{cases} \text{equation}_1 & \text{if condition}_1 \\ \text{equation}_2 & \text{if condition}_2 \\ \vdots & \vdots \\ \text{equation}_n & \text{if condition}_n \end{cases}

Examples

Section 2

Determining Domain Conditions for Piecewise Functions

Property

For a piecewise function f(x)={f1(x)if x satisfies condition 1f2(x)if x satisfies condition 2⋮⋮f(x) = \begin{cases} f_1(x) & \text{if } x \text{ satisfies condition 1} \\ f_2(x) & \text{if } x \text{ satisfies condition 2} \\ \vdots & \vdots \end{cases}, determine which piece applies by checking which condition the input value satisfies.

Examples

Section 3

Evaluating Piecewise Functions

Property

To evaluate a piecewise function at a given input value, first determine which piece (or condition) the input satisfies, then substitute the input into the corresponding function rule for that piece.

Examples

Section 4

Graphing Piecewise Functions

Property

To graph a piecewise function, we plot each piece separately on its specified domain, then combine them into one graph. Each piece is graphed only over its defined interval, and we use open or closed circles to indicate whether endpoints are included.

Examples