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Updated Feb 24, 2026

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8 pages

Fun with Piecewise Functions: Easy Examples and How to Solve Them

Shaivi Ramisetti

@shaivi711

A comprehensive guide to understanding and evaluating piecewise functions,... Show more

1 / 8

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 2: Evaluation Techniques

This page demonstrates piecewise functions evaluation examples through practical problems. It shows how to determine which piece of the function to use based on the input value.

Highlight: When evaluating piecewise functions, first identify which interval contains your input value, then use the corresponding function piece.

Example: For f(-5), since -5 < -3, use the first piece: 2x-5 Solution: 2(-5)-5 = -10-5 = -15

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 3: Graphical Analysis

This page presents a graphical representation of a piecewise function, emphasizing the visual interpretation of function behavior.

Highlight: The graph shows distinct pieces connecting at transition points, illustrating the continuous or discontinuous nature of the function.

Example: Finding f(0) = 2 demonstrates how to read function values directly from the graph.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 4: Domain Restrictions

This page explores piecewise functions with specific domain restrictions and their graphical representations.

Vocabulary: Domain restrictions define the intervals where each piece of the function is valid.

Example: The function transitions at x = -1 and x = 3, creating distinct regions of behavior.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 5: Linear Piecewise Functions

This page focuses on piecewise functions composed of linear pieces with different slopes and y-intercepts.

Example: The function consists of three pieces:

f(x) = 2x+5 for x ≤ 0
f(x) = -2x+12 for 0 < x ≤ 4
A constant value for x > 4

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 6: Step Functions

This page introduces step functions as a special case of piecewise functions, showing discontinuous behavior at transition points.

Definition: A step function is a piecewise function that has constant values over specific intervals.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 7: Point Functions

This page demonstrates piecewise functions defined at specific points, including isolated points and intervals.

Highlight: Some piecewise functions may include individual points with distinct values, creating discontinuities.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 8: Quadratic Pieces

This page explores piecewise functions containing quadratic expressions, showing how parabolic sections can be combined with other function types.

Example: The function includes a quadratic piece defined over the interval -2 < x < 2, demonstrating how curved sections can be incorporated into piecewise functions.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Page 1: Introduction to Piecewise Functions

This page introduces the fundamental concept of piecewise functions. A piecewise function consists of multiple sub-functions, each defined over specific intervals of the domain.

Definition: A piecewise function is a mathematical function composed of multiple sub-functions, where each piece applies to a specific interval of the input values.

Example: The function presented shows three distinct pieces:

f(x) = 2x-5 for x < -3
f(x) = -x² + 4x for -3 < x < 7
f(x) = √2x+6 for x ≥ 7

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Pre-Calculus

•

Updated Feb 24, 2026

•

8 pages

Fun with Piecewise Functions: Easy Examples and How to Solve Them

Shaivi Ramisetti

@shaivi711

A comprehensive guide to understanding and evaluating piecewise functions, featuring multiple examples and detailed explanations of function behavior across different domains.

Piecewise functions consist of multiple sub-functions defined over specific intervals
Each piece of the function operates within its... Show more

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

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Page 2: Evaluation Techniques

This page demonstrates piecewise functions evaluation examples through practical problems. It shows how to determine which piece of the function to use based on the input value.

Highlight: When evaluating piecewise functions, first identify which interval contains your input value, then use the corresponding function piece.

Example: For f(-5), since -5 < -3, use the first piece: 2x-5 Solution: 2(-5)-5 = -10-5 = -15

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Page 3: Graphical Analysis

This page presents a graphical representation of a piecewise function, emphasizing the visual interpretation of function behavior.

Highlight: The graph shows distinct pieces connecting at transition points, illustrating the continuous or discontinuous nature of the function.

Example: Finding f(0) = 2 demonstrates how to read function values directly from the graph.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Page 4: Domain Restrictions

This page explores piecewise functions with specific domain restrictions and their graphical representations.

Vocabulary: Domain restrictions define the intervals where each piece of the function is valid.

Example: The function transitions at x = -1 and x = 3, creating distinct regions of behavior.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Sign up to see the contentIt's free!

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Page 5: Linear Piecewise Functions

This page focuses on piecewise functions composed of linear pieces with different slopes and y-intercepts.

Example: The function consists of three pieces:

f(x) = 2x+5 for x ≤ 0
f(x) = -2x+12 for 0 < x ≤ 4
A constant value for x > 4

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Sign up to see the contentIt's free!

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Page 6: Step Functions

This page introduces step functions as a special case of piecewise functions, showing discontinuous behavior at transition points.

Definition: A step function is a piecewise function that has constant values over specific intervals.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

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Page 7: Point Functions

This page demonstrates piecewise functions defined at specific points, including isolated points and intervals.

Highlight: Some piecewise functions may include individual points with distinct values, creating discontinuities.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

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Page 8: Quadratic Pieces

This page explores piecewise functions containing quadratic expressions, showing how parabolic sections can be combined with other function types.

Example: The function includes a quadratic piece defined over the interval -2 < x < 2, demonstrating how curved sections can be incorporated into piecewise functions.

$# PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. $f(x) = \begin{cases} 2|x-5| & \text{if } x < -3 \\ -x^2 +$

Sign up to see the contentIt's free!

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Page 1: Introduction to Piecewise Functions

This page introduces the fundamental concept of piecewise functions. A piecewise function consists of multiple sub-functions, each defined over specific intervals of the domain.

Definition: A piecewise function is a mathematical function composed of multiple sub-functions, where each piece applies to a specific interval of the input values.

Example: The function presented shows three distinct pieces:

f(x) = 2x-5 for x < -3
f(x) = -x² + 4x for -3 < x < 7
f(x) = √2x+6 for x ≥ 7