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Composing Functions

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

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

@shaivi711

·

74 Followers

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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 own domain, requiring careful attention to interval boundaries
  • Evaluation involves selecting the correct function piece based on the input value
  • Visual representations through graphs help understand function behavior across different domains
  • Practice problems demonstrate how to evaluate piecewise functions through step-by-step solutions

6/21/2023

59

PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

View

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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Subjects

/

Pre-Calculus

/

Functions

/

Composing Functions

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

user profile picture

Shaivi Ramisetti

@shaivi711

·

74 Followers

Follow

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 own domain, requiring careful attention to interval boundaries
  • Evaluation involves selecting the correct function piece based on the input value
  • Visual representations through graphs help understand function behavior across different domains
  • Practice problems demonstrate how to evaluate piecewise functions through step-by-step solutions

6/21/2023

59

 

10th/11th

 

Pre-Calculus

3

PIECEWISE FUNCTIONS Made up of 2 or more functions, each with its own domain. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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. 2x-5 f(x) = {-x² + 4x √√2x+6 if x <-3 if-3 < x <7 if x ≥7 EVAL

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

Similar Content

Pre-Calculus - 1.08 Piecewise functions

What is a piecewise function and how to solve it.

1

50

0

Know 1.08 Piecewise functions thumbnail

Pre-Calculus - Everything you need to know for Functions

This notes have everything you need to understand and know about functions in Pre-Calc Chapter 1

8

244

0

Know Everything you need to know for Functions thumbnail

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

15 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying