inverses of linear functions algebra 2 homework answers

Inverse Functions Worksheet

Students will practice work with inverse functions including identifying the inverse functions , graphing inverses and more.

Example Questions

Directions: Find the inverse of each function

Challenge Problems. Part II

• Inverse Function
• Relations and Functions -- everything you might want to know.
• Domain and Range
• Functions and Relations in Math

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

Infinite Algebra 2

Test and worksheet generator for algebra 2.

Infinite Algebra 2 covers all typical Algebra 2 material, beginning with a few major Algebra 1 concepts and going through trigonometry. There are over 125 topics in all, from multi-step equations to trigonometric identities. Suitable for any class with advanced algebra content. Designed for all levels of learners, from remedial to advanced.

Included in version 2.70.06 released 1/5/2023:

• Fixed: Choices could appear incorrectly

Included in version 2.70.05 released 12/21/2022:

• Fixed: Continuous Exponential Growth and Decay Word Problems - Could generate an incorrect answer

Included in version 2.70 released 7/8/2022:

• New: Add preferences for Metric/Imperial units
• New: Add preference to “Prefer x“ as the variable letter
• New topic: Discrete Relations
• New topic: Continuous Relations
• New topic: Evaluating and Graphing Functions
• New topic: Direct and Inverse Variation
• New topic: Continuous Exponential Growth and Decay
• Improved: UI, security, and stability with updated libraries
• Fixed: Properties of Exponents – Choices could be equivalent
• Fixed: Properties of Logarithms – Choices could be equivalent
• Fixed: Solving Exponential Equations With Logarithms – Choices could be equivalent
• Fixed: Geometric Series – Choices could be equivalent

Included in version 2.62 released 2/8/2022:

• New: Print questions from Presentation View
• New Topic: Discrete Exponential Growth and Decay Word Problems
• Improved: UI, security, and stability with updated framework and libraries
• Improved: [Mac] Dark mode
• Fixed: [Windows] - Detached topic list does not close when application is closed
• Fixed: Probability Mutually Exclusive Word Problems - mutually exclusive case given when inappropriate

Included in version 2.61.03 released 9/10/2021:

• Improved: Minor UI improvements
• Fixed: Law of Cosines - Crash when opening topic
• Fixed: [Windows] Unable to change page layout to landscape
• Fixed: [Windows] Error 'Unable to load platform plugin'

Included in version 2.61 released 8/27/2021:

• Fixed: Preferences page for Kuta Works
• Improved: Security and stability with updated networking libraries

Included in version 2.60 released 8/19/2021:

• Improved: Site licenses check expiration date more frequently
• Fixed: Combinatoric Word Problems - Values could overflow
• Fixed: Properties of Exponents - Answer might not be fully simplified
• Fixed: Properties of Logs - Possible to have duplicate choices
• Fixed: Radical Equations - Option to mix radicals and rational exponents had no effect

Included in version 2.52 released 6/14/2019:

• New: Scramble questions by directions
• New: Scramble all questions in assignment
• New: Consolidate question sets with identical options
• New: Regroup question sets by directions
• New: Kuta Works - Create a two semester course
• New: Kuta Works - Clone assignments from a previous course into a new one
• Fixed: Presentation View - Arrow keys could change zoom at start/end of assignment
• Fixed: Tight Layout - Directions could be cramped for no reason
• Fixed: Inverse Trig - Angles limited for tan( x )
• Fixed: Inverse Trig - Errors in UI logic

Included in version 2.50 released 4/12/2019:

• New: Kuta Works - Option to hide answers and results from students until after due date
• New: Kuta Works - Option to control how long choices are hidden
• Improved: Options windows appear at a better initial size
• Improved: Window controls shown at their native size and spacing
• Improved: Network proxy configuration window
• Improved: License activation process
• Fixed: Presentation View - Answers could be cut off
• Fixed: Some window controls cut off on Windows 10 with display scaling
• Fixed: Spelling errors
• Fixed: Kuta Works - Course list could display more than just your active courses
• Fixed: Graphing Rational Functions - Open holes in graph could appear as filled holes

Included in version 2.42 released 12/11/2018:

• Fixed: Certain symbols incorrect if saved on Mac and loaded on Windows or vice versa
• Fixed: General Sequences - Recursive formula sometimes wrong
• Fixed: Properties of Logarithms - Avoid questions with reducible roots

Included in version 2.41 released 9/26/2018:

• Fixed: Archived courses in Kuta Works are now hidden
• Fixed: Generating or regenerating questions could cause a crash

Included in version 2.40 released 8/8/2018:

• New: Integrated with Kuta Works . Now post assignments online.

Included in version 2.25 released 4/27/2018:

• Improved: Generate questions more quickly
• Improved: Measurement arrows reach correct endpoints
• Improved: Absolute Value Equations - Better decimal numbers
• Improved: Absolute Value Inequalities - Better decimal numbers
• Improved: Compound Inequalities - Better decimal numbers
• Improved: Multi-Step Equations - Better decimal numbers and more predictable special cases
• Improved: Multi-Step Inequalities - Better decimal numbers
• Improved: Order of Operations - Better decimal numbers
• Fixed: Trig units for piecewise functions and ordered pairs
• Fixed: Graphing Rational Functions - Prevent equivalent choices
• Fixed: Law of Sines - Program could freeze
• Fixed: Transformations - Prevent equivalent choices

Included in version 2.18 released 7/31/2017:

• Fixed: Open circles on graphs were filled in when displayed as answer in red
• Fixed: Typo in sample custom question
• Fixed: Transformations - Program could freeze
• Fixed: Systems of Quadratic Equations - Program could freeze

Included in version 2.17 released 4/27/2017:

• Improved: [Windows] "Export to clipboard as bitmap" renders image with better quality
• Fixed: Probability with Combinations and Permutations - Could ask about impossible lottery ball

Included in version 2.16.20 released 11/16/2016:

• Fixed: Solving Quadratic Equations by Taking Square Roots - Option to "Allow fractions" not working as expected

Included in version 2.16 released 9/16/2016:

• Fixed: Numbers with many significant digits could round incorrectly
• Fixed: Customized question containing an error could not be modified
• Fixed: Customized question containing an error could cause a crash
• Fixed: Feedback tool detects Windows 10 properly
• Fixed: Mouse cursor when drag-dropping a question
• Fixed: Keyboard shortcut for changing order of question sets in assignment

Included in version 2.15 released 7/14/2016:

• Improved: Reorder question sets by dragging and dropping
• Improved: Move individual questions within a set by dragging and dropping
• Improved: Presentation View - Commands now on a movable, dockable toolbar
• Improved: Added option for large toolbar icons
• Improved: New icons
• Fixed: Certain options could result in double parenthesis or parenthesis around a negative sign
• Fixed: Arrow over vector variable name not red when appropriate
• Fixed: Solving Systems of Equations by Graphing - Answer could be displayed twice
• Fixed: Equations of Parabolas - Certain givens indicate wrong sign

Included in version 2.11 released 5/13/2016:

• Improved: [Windows] "Export to clipboard as bitmap" pastes into Microsoft Word at appropriate DPI
• Fixed: Keyboard shortcut for adding space to question set

Included in version 2.10 released 5/11/2016:

• New: Preference for including "None of the above" as a choice
• New: One column layout
• New: Added support for Infinite Precalculus
• New: Easily add piecewise functions of graphs in custom questions: Example: piecewise([2x-3] if [x<5], [x-1] if [x >= 5])
• New: Easily add functions with restricted domains to graphs in custom questions Example: function(x/2, x<0)
• Improved: More efficient layout of choices
• Improved: [Mac] Added option for legal paper in page setup
• Fixed: Spelling and grammatical errors
• Fixed: Graphing Quadratic Functions - Prevent equivalent choices
• Fixed: Binomial Theorem - Certain options didn't allow for many questions
• Fixed: Right Triangle Trig - Couldn't select just one ratio except sine
• Fixed: Right Triangle Trig - Answer sometimes given in problem
• Fixed: Law of Cosines - Certain SAS cases could freeze program
• Note: Older versions will not be able to open assignments saved with this version

Included in version 2.06 released 8/3/2015:

• Fixed: Crash when saving if previously used folder no longer exists

Included in version 2.05 released 7/20/2015:

• New Topic: Sample Spaces and The Fundamental Counting Principle
• New Topic: Independent and Dependent Events
• New Topic: Independent and Dependent Events Word Problems
• New Topic: Mutually Exclusive Events
• New Topic: Mutually Exclusive Events Word Problems
• Fixed: Classifying Conics - Checkboxes for conic types had no effect
• Fixed: Presentation View - Displayed wrong directions in 3-up mode
• Fixed: Presentation View - Lines too long after creating new problems
• Fixed: [Windows] Uncommon error when using the network

Included in version 2.04.40 released 3/31/2015:

• Fixed: Activation data unrecognized under special circumstances
• Fixed: [Windows] Help files broken by previous release
• Fixed: Order of Operations - Radio buttons reversed

Included in version 2.04.20 released 3/23/2015:

• Improved: [Windows] "Export to clipboard as bitmap" uses a transparent background

Included in version 2.04 released 3/17/2015:

• New Topic: Permutations
• New Topic: Combinations
• New Topic: Permutations vs Combinations
• New Topic: Probability with Permutations and Combinations
• New Topic: Converting Degrees and Degrees-Minutes-Seconds
• [factorial(n)]
• [perm(n, r)]
• [comb(n, r)]
• Improved: Better support for proxies
• Improved: Support for HiDPI Mac Retina displays
• Improved: Print options window - Some settings are now persistent
• Improved: Presentation View - Use arrow keys to navigate between questions
• Improved: Question set updates free response/multiple-choice when all questions have changed
• Improved: [Mac] Activation also checks /Library/Preferences/ks-config.txt for serial numbers
• Fixed: [Windows] Crash when printing with certain printers
• Fixed: Crash when loading an assignment with over 80 custom questions
• Fixed: Custom question answer was never choice A
• Fixed: Paper size set before printing
• Fixed: Cramer's Rule - Fixed directions
• Fixed: Systems of Three Equations - Restored the option to "only solve for x"
• Fixed: Center of circle wasn't appearing

Included in version 2.03 released 9/19/2014:

• Fixed: Preference for question layout mode not correctly loaded
• Fixed: Issues with automatic spacing and highlighting questions when using tight layout
• Fixed: In Presentation View, graphing problems display choices more compactly
• Fixed: Printing blank pages on HP LaserJet 1018, 1020, and 1022 among others
• Fixed: Logs and Exponents as Inverses - problem with slider

Included in version 2.02 released 8/5/2014:

• New: Software uses system proxy when necessary
• Improved: Activation window has taskbar entry
• Improved: Remembers the last directory used when saving and opening files
• Fixed: [Windows only] Possible crash when loading an assignment with a lot of custom questions
• Fixed: [Mac only] Application stops responding to hover events
• Fixed: Activation from before v2.0 sometimes not recognized
• Fixed: Export to clipboard sometimes cuts off choices B and D
• Fixed: Help contents could become unusable
• Fixed: Word wrap when modifying a question
• Fixed: Memory leak

Included in version 2.01 released 7/28/2014:

• Fixed: Could crash when opening an assignment from a different program

Included in version 2.00 released 7/23/2014:

• New: Print to PDF
• New: Filter for topic list
• New: Zoom while viewing on screen
• New: "More like these" button in presentation mode
• New: Show where punch holes will go on the page
• New: Print only an answer sheet
• New: When printing, page range is enabled
• New: Page elements are outlined when hovered
• New: Double-click on directions to change them
• New: When merging assignments, options to put similar topics together or put them end-to-end
• New: Highlight and go to the questions in a question set (menu command or shift-click)
• New: Site licenses can be activated per-machine if run by the administrator
• Ctrl-Click on a question
• Shift-Click on a question
• Ctrl-Shift-Click on a question
• Shift-Click in a empty area of a page
• Ctrl-Shift-Click in a empty area of a page
• Shift-Click on a question set
• Ctrl-Shift-Click on a question set
• Improved: Assignment files are drastically smaller
• Improved: Sidebar is dockable on either side, can be floated, and position reset
• Improved: Auto-spacing doesn't leave dead space at bottom of each page
• Improved: Preview when editing directions
• Improved: Keyboard shortcuts work no matter where the focus is
• Improved: Scale number of questions window more user-friendly
• Fixed: Minor bugs in seven topics

Included in version 1.56 released 8/14/2013:

• Fixed: Graphing Exponential Functions - Correct choice used lighter shade of blue
• Fixed: Radical Equations could be missing a solution
• Fixed: Graphs of Polynomial Functions - Inflection points sometimes listed as local extrema
• Fixed: Properties of Exponents - Occasionally answer wasn't fully simplified
• Fixed: Dividing Polynomials - Occasionally polynomial was evenly divisible when it shouldn't have been
• Fixed: Graphing Rational Functions - Minor UI issue
• Fixed: Geometric Sequences - Given information sometimes allowed for two possible common ratios
• Fixed: Dividing Polynomials could freeze
• Fixed: Systems of Equations by Elimination - "No solution" occasionally should have been "Infinite solutions"

Included in version 1.55 released 12/11/2012:

• New: Graphs can be added to custom questions
• Improved: Graphing and Graph Paper utility more powerful and easier to use
• Improved: Support for loading files from Infinite Calculus
• Improved: Faster save/load
• Fixed: Answer for Factoring Quadratic Expressions sometimes incorrect
• Fixed: Choices for Evaluating Functions with a variable operand could contain wrong variable
• Fixed: Choices for Function Operations with a variable operand could contain wrong variable
• Fixed: Function Operations, functions when composed with self could provide the wrong answer
• Fixed: Custom questions with an illegal expression could freeze the program
• Fixed: Certain families of functions graphed incorrectly

Included in version 1.53 released 9/11/2012:

• New: Preference for notation for greatest integer function
• New: Added maximize/restore button to Presentation View
• New: Graphing Absolute Value Equations has options for including coefficients
• Improved: Help files
• Improved: Scroll bars
• Improved: User interface
• Improved: When choices make a question too tall for a page, some choices are removed
• Improved: Algebraic simplification routines are now more efficient
• Improved: Better graphs for: Graphing Absolute Value Equations, Graphing Linear Equations, Graphing Exponential Functions, Graphing Quadratic Functions, Graphing Linear Inequalities, Graphing Systems of Linear Inequalities
• Improved: Better number lines for: Multi-Step Inequalities, Compound Inequalities, Absolute Value Inequalities
• Fixed: Wrapping to full-page and half-page could be too wide
• Fixed: Able to graph certain families of functions
• Fixed: Issue involving loading & regenerating an assignment and answers being hidden
• Fixed: Program could ask about changing directions too much
• Fixed: Graphs for Compound Inequalities could omit interval
• Fixed: Double & Half Angle Identities: Button to show more examples did not work
• Fixed: Writing Linear Equations: Answer can't be line given in question

Included in version 1.52 released 5/29/2012:

• Improved: More professional radical signs
• Improved: Graphs of discontinuous functions have breaks and open / closed holes
• Improved: Even less likely to crash when generating questions
• Fixed: Word wrapping could skip blank lines, not wrap where appropriate
• Fixed: Wrapping to full page in a custom question was too wide
• Fixed: Products of powers of e (like 2 e &sup3;) no longer display with a multiplication dot
• Fixed: Most diagrams appeared incorrectly in print preview
• Fixed: Random error message on save or load
• Fixed: Graphing Rational Functions: Open holes in graph could appear as filled holes

Included in version 1.51.02 released on 4/9/2012:

• Fixed: Minimum Windows version was set incorrectly

Included in version 1.51 released 4/5/2012:

• New: Added greatest integer function / floor function to custom questions    Example: Find [int(2x)] when [x=3/5]
• New: Added piecewise functions to custom questions    Example: [eval(f,x)] = [piecewise [x] if [x < 0], [xx] if [x >= 0]]
• Improved: Faster! Optimized rendering of questions to screen
• Improved: Faster! Optimized graphing routines
• Improved: Improved graphing capabilities
• Improved: Diagrams drawn more smoothly on screen
• Fixed: Rewriting Logarithms - Numeric problems sometimes had bad values
• Fixed: Evaluating Logarithms - Problems sometimes had bad values
• Fixed: Graphs could omit holes
• Fixed: Graphs of constant functions or those involving e and π could be incorrect
• Fixed: Minor indentation issue in custom questions
• Fixed: Horizontal asymptotes could be drawn beyond a graph's area

Included in version 1.50 released 3/15/2012:

• New: Presentation View window is resizeable
• New: Presentation View has option to automatically hide the answers when a new question is displayed
• New: High-level filter to prevent questions from containing an illegal expression
• Improved: Function notation can now be edited
• Improved: Faster! Optimized the simplification of mathematical expressions
• Improved: Faster! Improved undo/redo algorithm
• Improved: Smaller executable size
• Improved: "Current Question Sets" list easier to use
• Improved: Options windows resize more smoothly
• Fixed: Equation of Parabolas could freeze
• Fixed: Geometric Series questions could list "Illegal Expression" as a choice
• Fixed: Graphing Exponential Functions could freeze
• Fixed: Systems of Equations Word Problems could freeze
• Fixed: Punctuation in some word problems
• Fixed: Certain expressions in a custom question would cause the software to crash
• Fixed: Expressions like root × term would not print a multiplication dot
• Note: Beginning with this version, Windows XP SP3 is the minimum required version of Windows

Included in version 1.45 released 4/12/2011:

• New: Assignments from this program can be opened and modified by our other programs          (Assignments saved with v1.45 or greater can be opened by other programs v1.45 or greater)
• New: Student data fields (name, date, period) can be renamed
• New: Additional student data field available
• New: Add & Continue is available when modifying an existing question set
• New: Link to this details page when a software update is available
• New: Product serial numbers can be placed in config.txt to facilitate enterprise installations
• Improved: Better backwards- and future-compatibility
• Fixed: Assignments with quotes in the title don't prevent Save As
• Fixed: Quotients could be sometimes be simplified incorrectly
• Fixed: Choices for dilations sometimes inappropriate
• Fixed: Dilations rounding x- and y-coordinates
• Fixed: Function Operations could have "Illegal Expression" as an answer or as a choice
• Fixed: Multi-Step Equations: Program could freeze
• Fixed: Dividing Radical Expressions: Program could freeze
• Fixed: Add & Continue respects change in problem type
• Fixed: Powers of i are correctly displayed

Included in version 1.42 released on 6/28/2010:

• New: Automatically checks for updates
• New: List topics by index order or suggested order

Included in version 1.40 released 2/8/2010:

• New Preference: 'Spacious' or 'tight' layout of questions on page
• Improved: Question sets with one question are more intelligently spaced
• Improved: Graphs with small ranges now look better
• Improved: Options screen now resizeable
• Fixed: Directions occasionally orphaned at bottom of page
• Fixed: Correct numbers changed because zeros at the end of a number were deleted (50 --> 5)

Included in version 1.08.20 released 5/22/2009:

• Changed: Minor change to license agreement so that renewals extend the termination date instead of beginning a new term.

Included in version 1.08 released 4/10/2009:

• New: Possible to change directions for a set without regenerating questions
• Improved: Changing directions cascades changes to appropriate neighbors
• Correction: Answers to Properties of Exponents could have wrong sign
• Correction: Inequalities were sometimes solved incorrectly
• Fixed: Software sometimes failed to renumber questions in assignment

Included in version 1.07 released 1/6/2009:

• New: Trig. equations that don't require factoring
• Correction: Function operations / linear combinations answers sometimes wrong
• Fixed: Software crashes when starting if closed when side panel is minimized

Included in version 1.06 released 12/4/2008:

• Improved: Isolate up to four questions on screen at once
• Improved: Support for computers with multiple processors
• Improved: User interface: Toolbar + Preferences
• Fixed: Obscure error involving selecting a question and scaling the assignment

Included in version 1.03 released 10/1/2008:

• New: Export individual questions as bitmaps
• Improved: Custom questions wrap to whole/half page
• Improved: Insert special symbols and math in custom questions
• Improved: Insert 'blanks' into math text
• Improved: Line thickness preference also now controls darkness of grid lines in graphs
• Correction: Writing logs in terms of others sometimes created duplicate questions
• Fixed: Internet calls would sometimes report wrong error

Included in version 1.02 released 9/18/2008:

• Correction: Question type was not saving all of its options
• Correction: Equations of Hyperbolas description was sometimes wrong
• Correction: Geometric Series directions was sometimes wrong
• Correction: Binomial Theorem type could cause the program to freeze
• Correction: Quadratic/Linear systems sometimes had the wrong answer
• Correction: Rational Exponents answers now correct
• Fixed: Internet calls could freeze program

Try Infinite Algebra 2 Today!

Discover the power and flexibility of our software firsthand with a free, 14-day trial. Installation is fast and simple. Within minutes, you can have the software installed and create the precise worksheets you need -- even for today's lesson.

Use each trial for up to 14 days. The trial version is identical to the retail version except that you cannot print to electronic formats such as PDF.

Mrs. Snow's Math

Mcneil high school.

• Parent Letter
•  Algebra I
• Math Modeling

Algebra II Lesson Notes

These notes follow the Prentice Hall Algebra II Texas Edition Textbook.   Roundrock ISD adopted new math textbooks to be used starting with the 2015 school year. 

I have kept these notes available for parents and students alike as basic algebraic fundamentals do not change.

FALL SEMESTER

Lesson 1 Parent Functions

Class Notes

Parent Functions Worksheet  

Lesson 2 Domain and Range

Domain and Range Worksheet  

Lesson 3 Function Transformations

Function Transformations Worksheet  

Lesson 4 Function Notation and Function Representations

Function Notation and Representations Worksheet  

Lesson 5 Inverse Functions

Inverse Functions Worksheet

Test Review (fall 2014)

Review Answers (fall 2014)

Test Review (fall 2013)

Review Answers   (fall 2013)

Chapter 1    Tools of Algebra

Lesson 1 Part 1   Properties of Real Numbers

Lesson 1 Part 2    

Lesson 2   Algebraic Expressions

Lesson 3   Solving Equations

Lesson 4    Solving Inequalities

Lesson 5    Absolute Value Equations and Inequalities

Lesson 6    Probability

Chapter 1 Rev .    (fall 2013)

Chapter 2    Functions, Equations And Graphs  

Lesson 1   Relations and Functions

Lesson 2   Linear equations  

Lesson 3   Direct Variation

Lesson 4   Using Linear Models

Lesson 5   Absolute Value Functions and Graphs

Lesson 2.5 Absolute Value Equations and Functions 2014

Worksheet to accompany part 1

Lesson 2.5 Absolute Value Transformations 2014

2014 Absolute Value Test Review

2014 A Review Solutions

Lesson 6 Families of Functions

Lesson 7   Two-Variable Inequalities

Class Notes Linear Inequalities

Class Notes Absolute Value Inequalities (Combined w/ 2.6)

2013 Absolute Value Test Review

2013 Absolute Value Review Solutions

Chapter 2 Rev. (review from a previous year)

Review Solutions 

Using Stat Plot

  Absolute Value Translations 

Chapter 3  Linear Systems

Lesson 1   Graphing Systems of Equations

Lesson 2  Solving Systems Algebraically

Lesson 3   Systems of Inequalities

Lesson 4   Linear Programming

Worksheet Assignment

Lesson 6   Systems with Three Variables

Chapter2&3 Review

Linear systems Review - 2.2, 2.7 and Chapter 3

Linear Systems Review - covering sections from Chapters 2 and 3

Chapter 3 Rev. (review from a previous Year)

Chapter 3 Rev. Extras

Chapter 4  Matrices

Lesson 1   Organizing Data into Matrices (2013 notes)

Lesson 2    Adding and Subtracting Matrices

Lesson 3    Matrix Multiplication

4.2 and 4.3 Class Notes  

Lesson 5 2x2 Matrices, Determinants and Inverses

  4.5 Class Notes

Solving Systems of 3 Equations : 

         by hand and with Gaussian Elimination

Homework Worksheet

Lesson 4.6 3x3 Matrices, Determinants and Inverses

and calculator methods to solve matrices

Class Notes -  3x3 Matrices, Determinants and Inverses

Chapter 4 Matrix Test Review (2014)

Review Solutions (2014)

Lesson 5&6 Det.   Determinants (2009)

Lesson 5&6 Inv.   Inverses (2009)

Lesson 3.6 Systems of Equations with 3 Variables

Lesson 7   Inverse Matrices and Systems (2013 notes)

Class Notes - 4.6 and 4.7 combined

Lesson 8   Augmented Matrices and Systems (2013 notes)

Chapter 4 Rev.    (2009 review)

Matrices Review - 2013  

Solutions   (fall 2013)

Chapter 5  Quadratic Equations and Functions

Last word on Chapt. 4

Lesson 1   Modeling Data with Quadratic Functions

Lesson 2   Properties of Parabolas

Class Notes for 5.1 and 5.2

Lesson 3   Transforming Parabolas

Lesson 4   Factoring Quadratic Expressions

Factoring Flow Chart

2013Chapter 5.1-5.4 Review 2013

Review Solutions   (fall 2013)

Lesson 5   Quadratic Equations

Lesson 8 Quadratic Formula

Class Notes for 5.5 and 5.8

Radicals  

Lesson 6   Complex Numbers

Lesson 7   Completing the Square

Solving Quadratic Inequalities

Worksheet Solving Quadratic Inequalities and Quadratic Word Problems

Chapter 5B Review   (fall 2014)

Solutions   (fall 2014)

Archive review  

Chapter 5 Review Part 1 2010

Chapter 5 review Part 2  2010

Fall Final 2014

Fall Final 2013

Solutions  

Fall 2010 Review :

Fall Final Review

SPRING SEMESTER

C hapter 6   Polynomials and Polynomial Functions

Lesson 6-1   Polynomial Functions

Class Notes  

Lesson 6-2   Polynomials and Linear Factors

Lesson 6-3   Dividing Polynomials

Lesson 6-4   Solving Polynomial Equations

ClassNotes  

Lesson 6-5/Lesson 6.6 Part I  Theorems about Polynomial Functions

Lesson 6- 5/ Lesson 6.6 Part II Finding Roots of Cubic Functions

Lesson 6-8   The Binomial Theorem

Ch 6 Review 2015

Ch 6 Review Solutions 2015

Ch 6 Review 2011

Chapter 7     Radical Functions and Rational Exponents

Lesson 7-8 Graphs of Square Root Functions

Class Notes (2015)

Lesson 7-1   Roots and Radical Expressions

Lesson 7-2   Multiplying and Dividing Radical Expressions

Class Notes 7-1 and 7-2

Lesson 7-3  Binomial Radical Expressions

Lesson 7-4   Rational Exponents

Lesson 7-5   Solving Square Root and Other Radical Equations

Lesson 7-6   Function Operations

Lesson 7-7    Inverse Relations and Functions

Lesson 7-8   Graphing Square Root and Other Radical Function

Chapter 7 Review (Spring 2015)

Chapter 7 Solutions (Spring 2015)

Lesson 7.1-7.4 Review  2011

  Ch. 7  Review 2011

Chapter 8  Exponential and logarithmic Functions

Lesson 8-1   Exploring Exponential Models

Lesson 8-2   Models of Exponential Functions

Worksheet Models of Exponential Functions

Lesson 8.2- B Graphs of Exponential Functions

Worksheet Graphs of Exponential Functions

Ch 8.1-8.2Review (Spring 2015)

Solutions (Spring 2015)

Ch.8-a and Ch.7 Spiral Review 2014

Lesson 8-3   Logarithmic Functions as Inverses

Lesson 8-4   Properties of Logarithms

Lesson 8-5   Exponential and Logarithmic Equations

Examples Pertaining to Logarithm Applications

Logarithm Applications Worksheet  

Lesson 8-6   Natural Logarithms

Chapter 8B Review 201 5

Solutions 201 5

Ch. 8 Review 2011

Chapter 9  Rational Functions

Lesson 9-1   Inverse Variation

Lesson 9-2   The Reciprocal Function Family

Lesson 9-3   Rational Functions and Their Graphs

Graphing Rational Functions Worksheet

Lesson 9-4   Rational Expressions

Rational Functions Test Review (2015)

Solutions (2015)

Lesson 9-5    Adding and Subtracting Rational Expressions

Lesson 9-6   Solving Rational Equations

Rational Equations Application Problems

Worksheet Rational Equation Applications

Chapter 9B 2015

Chapter 9B Solutions 2015

Ch. 9-A Review2014

2014 Review Solutions  

Chapter 9B Test Review 2014 9.4-9.6

Solutions 2014

Ch. 9 Review 2011

Chapter 10  Quadratic relations and Conic Sections

Lesson 10-3   Circles        

Class Notes                  

Lesson 10-4   Ellipses

Lesson 10-5   Hyperbolas

Lesson 10-2   Parabolas

Parabolas Homework Worksheet

Lesson 10-6   Translating Conic Sections

10.6 Homework Worksheet

Chapter 10 Review

Solutions 2015

Which Conics?  Flow Chart

Conics Formula Sheet

Completing the Square

Solving Conics Problems  

Ch. 10 Review (2011)

Spring 2015 Review

Review 2015

Spring 2014 Review

Spring 2014 Final Review

2014 Solutions

Copyright © 2009. All Rights Reserved.

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Inverse of Linear Functions

Catch-up and review.

Here are a few recommended readings before getting started with this lesson.

• Solving Multi-Step Equations
• Literal Equation
• Linear Function
• Slope-Intercept Form
• Graphing Linear Functions in Slope-Intercept Form
• Function Notation

Undoing a Function

Inverse relation, inverse of a function.

Every function has an inverse relation . If this inverse relation is also a function, then it is called an inverse function . In other words, the inverse of a function f is another function f^(- 1) such that they undo each other.

f(f^(- 1)(x))=x and f^(- 1)(f(x))=x

Also, if x is the input of a function f and y its corresponding output , then y is the input of f^(- 1) and x its corresponding output.

f(x)=y ⇔ f^(- 1)(y)=x

f^(- 1)(x)= x+3/2

Remove parentheses

Subtract term

Therefore, f and f^(- 1) undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of f^(- 1) are the reversed points on the graph of f.

Using Graphs to Determine if Two Functions Are Inverse Functions

Kriz enjoys playing video games with their friends. For their birthday they received a copy of Mathleaks: The Adventure, a math-based video game.

Are f and g Inverse Functions? Yes.

Graph both functions on the same coordinate plane and see if they are each other's reflection across the line y=x.

Since f and g are linear functions written in slope-intercept form , they can both be graphed using their slope and y-intercept .

Determining if Two Functions Are Inverses by Looking at Their Graphs

It was discussed previously that if the graphs of two functions are each other's reflection across y=x, then they are inverse functions . Therefore, it can be determined if two functions are inverse just by looking at their graphs. Determine whether the following linear functions are inverse by looking at the lines .

Verifying Inverse Functions by Composition

If Paulina gets an A on a math test, her mother will buy her a new saxophone.

To get an A, Paulina must answer two questions correctly. Help her get the new saxophone!

g(x)= 1/5x-2

Distribute 5

Associative Property of Multiplication

It has been found that both f(g(x)) and g(f(x)) are equal to x. Therefore, f and g are inverse functions.

h(x)=1/2x+6 and k(x)=2x-6 By following the same procedure as in Part A, the expressions for h(k(x)) and k(h(x)) will be found.

It has been found that neither h(k(x)) nor k(h(x)) is equal to x. Therefore, h and k are not inverse functions.

Determining if Two Functions Are Inverse by Composition

For the functions f and g, use a composition to determine whether they are inverse functions .

Finding the Inverse of a Function Algebraically

To begin, since f(x)=y describes the input-output relationship of the function, replace f(x) with y in the function rule. f(x)= 2x-1/3 → y= 2x-1/3

Because the inverse of a function reverses x and y, the variables can be switched. Notice that every other piece in the function rule remains the same. y= 2 x-1/3 switch x= 2 y-1/3

LHS * 3=RHS* 3

LHS+1=RHS+1

.LHS /2.=.RHS /2.

Rearrange equation

Just as f(x)=y shows the input-output relationship of f, so does f^(- 1)(x)=y. Therefore, replacing y with f^(- 1)(x) gives the rule for the inverse of f. y=3x+1/2 → f^(- 1)(x)=3x+1/2 Notice that in f, the input is multiplied by 2, decreased by 1, and divided by 3. From the rule of f^(- 1), it can be seen that x undergoes the inverse of these operations in the reverse order. Specifically, x is multiplied by 3, increased by 1, and divided by 2.

Finding the Inverse of a Function

Zosia lives in Honolulu, Hawaii.

Start by replacing f(x) with y. After that, switch the x- and y-variables. Then, solve the obtained equation for y. Finally, replace y with f^(- 1)(x).

To find the inverse of the function , there are four steps to follow.

• Replace f(x) with y.
• Switch x and y.
• Solve the obtained equation for y.
• Replace y with f^(- 1)(x).

These steps will be done one at a time.

Here, the variable y will be substituted for f(x) in the given function. f(x)=- 4/3x+24 → y=- 4/3x+24

In this step, the variables x and y must be switched. y=- 4/3 x+24 switch x=- 4/3 y+24

LHS-24=RHS-24

LHS * (- 3/4)=RHS* (- 3/4)

Distribute (- 3/4)

Commutative Property of Multiplication

a(- b)=- a * b

a*b/c= a* b/c

Calculate quotient

a-(- b)=a+b

Finally, f^(- 1)(x) will be substituted for the y-variable in the equation obtained in the previous step. y=- 3/4x+18 ↓ f^(- 1)(x)=- 3/4x+18

Finding the Inverse Function

Find the inverse of the given function . Write the answer in slope-intercept form — in the format y=mx+b. If the slope and y-intercept are not integers , express them as decimal numbers . If necessary, round them to two decimal places .

Start by replacing f(x) with y. After that, switch the x- and y-variables. Then, solve the obtained equation for y, and finally replace y with f^(- 1)(x).

Here, the variable y will be substituted for f(x) in the given function. f(x)=x+2 substitute y=x+2

In this step, the variables x and y must be switched. y= x+2 switch x= y+2

Here, the equation obtained in the previous step will be solved for y. x=y+2 ⇔ y=x-2

Recommended exercises

3.7 Inverse Functions

Learning objectives.

In this section, you will:

• Verify inverse functions.
• Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
• Find or evaluate the inverse of a function.
• Use the graph of a one-to-one function to graph its inverse function on the same axes.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.

If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. Figure 1 provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

Verifying That Two Functions Are Inverse Functions

Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula

and substitutes 75 for F F to calculate

Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the week’s weather forecast from Figure 2 for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.

At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for F F after substituting a value for C . C . For example, to convert 26 degrees Celsius, she could write

After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.

The formula for which Betty is searching corresponds to the idea of an inverse function , which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.

Given a function f ( x ) , f ( x ) , we represent its inverse as f − 1 ( x ) , f − 1 ( x ) , read as “ f “ f inverse of x . ” x . ” The raised −1 −1 is part of the notation. It is not an exponent; it does not imply a power of −1 −1 . In other words, f − 1 ( x ) f − 1 ( x ) does not mean 1 f ( x ) 1 f ( x ) because 1 f ( x ) 1 f ( x ) is the reciprocal of f f and not the inverse.

The “exponent-like” notation comes from an analogy between function composition and multiplication: just as a − 1 a = 1 a − 1 a = 1 (1 is the identity element for multiplication) for any nonzero number a , a , so f − 1 ∘ f f − 1 ∘ f equals the identity function, that is,

This holds for all x x in the domain of f . f . Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function f ( x ) , f ( x ) , we can verify whether some other function g ( x ) g ( x ) is the inverse of f ( x ) f ( x ) by checking if both g ( f ( x ) ) = x g ( f ( x ) ) = x and f ( g ( x ) ) = x f ( g ( x ) ) = x are true.

For example, y = 4 x y = 4 x and y = 1 4 x y = 1 4 x are inverse functions.

A few coordinate pairs from the graph of the function y = 4 x y = 4 x are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function y = 1 4 x y = 1 4 x are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

Inverse Function

For any one-to-one function f ( x ) = y , f ( x ) = y , a function f − 1 ( x ) f − 1 ( x ) is an inverse function of f f if f − 1 ( y ) = x . f − 1 ( y ) = x . This can also be written as f − 1 ( f ( x ) ) = x f − 1 ( f ( x ) ) = x for all x x in the domain of f . f . It also follows that f ( f − 1 ( x ) ) = x f ( f − 1 ( x ) ) = x for all x x in the domain of f − 1 f − 1 if f − 1 f − 1 is the inverse of f . f .

The notation f − 1 f − 1 is read “ f “ f inverse.” Like any other function, we can use any variable name as the input for f − 1 , f − 1 , so we will often write f − 1 ( x ) , f − 1 ( x ) , which we read as “ f “ f inverse of x . ” x . ” Keep in mind that

and not all functions have inverses.

Identifying an Inverse Function for a Given Input-Output Pair

If for a particular one-to-one function f ( 2 ) = 4 f ( 2 ) = 4 and f ( 5 ) = 12 , f ( 5 ) = 12 , what are the corresponding input and output values for the inverse function?

The inverse function reverses the input and output quantities, so if

Alternatively, if we want to name the inverse function g , g , then g ( 4 ) = 2 g ( 4 ) = 2 and g ( 12 ) = 5. g ( 12 ) = 5.

Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See Table 1 .

Given that h − 1 ( 6 ) = 2 , h − 1 ( 6 ) = 2 , what are the corresponding input and output values of the original function h ? h ?

Given two functions f ( x ) f ( x ) and g ( x ) , g ( x ) , test whether the functions are inverses of each other.

• Determine whether f ( g ( x ) ) = x f ( g ( x ) ) = x or g ( f ( x ) ) = x . g ( f ( x ) ) = x .
• If either statement is true, then both are true, and g = f − 1 g = f − 1 and f = g − 1 . f = g − 1 . If either statement is false, then both are false, and g ≠ f − 1 g ≠ f − 1 and f ≠ g − 1 . f ≠ g − 1 .

Testing Inverse Relationships Algebraically

If f ( x ) = 1 x + 2 f ( x ) = 1 x + 2 and g ( x ) = 1 x − 2 , g ( x ) = 1 x − 2 , is g = f − 1 ? g = f − 1 ?

We must also verify the other formula.

Notice the inverse operations are in reverse order of the operations from the original function.

If f ( x ) = x 3 − 4 f ( x ) = x 3 − 4 and g ( x ) = x + 4 3 , g ( x ) = x + 4 3 , is g = f − 1 ? g = f − 1 ?

Determining Inverse Relationships for Power Functions

If f ( x ) = x 3 f ( x ) = x 3 (the cube function) and g ( x ) = 1 3 x , g ( x ) = 1 3 x , is g = f − 1 ? g = f − 1 ?

No, the functions are not inverses.

The correct inverse to the cube is, of course, the cube root x 3 = x 1 3 , x 3 = x 1 3 , that is, the one-third is an exponent, not a multiplier.

If f ( x ) = ( x − 1 ) 3 and g ( x ) = x 3 + 1 , f ( x ) = ( x − 1 ) 3 and g ( x ) = x 3 + 1 , is g = f − 1 ? g = f − 1 ?

Finding Domain and Range of Inverse Functions

The outputs of the function f f are the inputs to f − 1 , f − 1 , so the range of f f is also the domain of f − 1 . f − 1 . Likewise, because the inputs to f f are the outputs of f − 1 , f − 1 , the domain of f f is the range of f − 1 . f − 1 . We can visualize the situation as in Figure 3 .

When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of f ( x ) = x f ( x ) = x is f − 1 ( x ) = x 2 , f − 1 ( x ) = x 2 , because a square “undoes” a square root; but the square is only the inverse of the square root on the domain [ 0 , ∞ ) , [ 0 , ∞ ) , since that is the range of f ( x ) = x . f ( x ) = x .

We can look at this problem from the other side, starting with the square (toolkit quadratic) function f ( x ) = x 2 . f ( x ) = x 2 . If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.

In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function f ( x ) = x 2 f ( x ) = x 2 with its domain limited to [ 0 , ∞ ) , [ 0 , ∞ ) , which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

If f ( x ) = ( x − 1 ) 2 f ( x ) = ( x − 1 ) 2 on [ 1 , ∞ ) , [ 1 , ∞ ) , then the inverse function is f − 1 ( x ) = x + 1. f − 1 ( x ) = x + 1.

• The domain of f f = range of f − 1 f − 1 = [ 1 , ∞ ) . [ 1 , ∞ ) .
• The domain of f − 1 f − 1 = range of f f = [ 0 , ∞ ) . [ 0 , ∞ ) .

Is it possible for a function to have more than one inverse?

No. If two supposedly different functions, say, g g and h , h , both meet the definition of being inverses of another function f , f , then you can prove that g = h . g = h . We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.

Domain and Range of Inverse Functions

The range of a function f ( x ) f ( x ) is the domain of the inverse function f − 1 ( x ) . f − 1 ( x ) .

The domain of f ( x ) f ( x ) is the range of f − 1 ( x ) . f − 1 ( x ) .

Given a function, find the domain and range of its inverse.

• If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.
• If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.

Finding the Inverses of Toolkit Functions

Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in Table 2 . We restrict the domain in such a fashion that the function assumes all y -values exactly once.

The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no inverse.

The absolute value function can be restricted to the domain [ 0 , ∞ ) , [ 0 , ∞ ) , where it is equal to the identity function.

The reciprocal-squared function can be restricted to the domain ( 0 , ∞ ) . ( 0 , ∞ ) .

We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure 4 . They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse.

The domain of function f f is ( 1 , ∞ ) ( 1 , ∞ ) and the range of function f f is ( −∞ , −2 ) . ( −∞ , −2 ) . Find the domain and range of the inverse function.

Finding and Evaluating Inverse Functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

Inverting Tabular Functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

Interpreting the Inverse of a Tabular Function

A function f ( t ) f ( t ) is given in Table 3 , showing distance in miles that a car has traveled in t t minutes. Find and interpret f − 1 ( 70 ) . f − 1 ( 70 ) .

The inverse function takes an output of f f and returns an input for f . f . So in the expression f − 1 ( 70 ) , f − 1 ( 70 ) , 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function f , f , 90 minutes, so f − 1 ( 70 ) = 90. f − 1 ( 70 ) = 90. The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if f ( a ) = b , f ( a ) = b , then f − 1 ( b ) = a . f − 1 ( b ) = a . By this definition, if we are given f − 1 ( 70 ) = a , f − 1 ( 70 ) = a , then we are looking for a value a a so that f ( a ) = 70. f ( a ) = 70. In this case, we are looking for a t t so that f ( t ) = 70 , f ( t ) = 70 , which is when t = 90. t = 90.

Using Table 4 , find and interpret ⓐ f ( 60 ) , f ( 60 ) , and ⓑ f − 1 ( 60 ) . f − 1 ( 60 ) .

Evaluating the Inverse of a Function, Given a Graph of the Original Function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

Given the graph of a function, evaluate its inverse at specific points.

• Find the desired input on the y -axis of the given graph.
• Read the inverse function’s output from the x -axis of the given graph.

Evaluating a Function and Its Inverse from a Graph at Specific Points

A function g ( x ) g ( x ) is given in Figure 5 . Find g ( 3 ) g ( 3 ) and g − 1 ( 3 ) . g − 1 ( 3 ) .

To evaluate g ( 3 ) , g ( 3 ) , we find 3 on the x -axis and find the corresponding output value on the y -axis. The point ( 3 , 1 ) ( 3 , 1 ) tells us that g ( 3 ) = 1. g ( 3 ) = 1.

To evaluate g − 1 ( 3 ) , g − 1 ( 3 ) , recall that by definition g − 1 ( 3 ) g − 1 ( 3 ) means the value of x for which g ( x ) = 3. g ( x ) = 3. By looking for the output value 3 on the vertical axis, we find the point ( 5 , 3 ) ( 5 , 3 ) on the graph, which means g ( 5 ) = 3 , g ( 5 ) = 3 , so by definition, g − 1 ( 3 ) = 5. g − 1 ( 3 ) = 5. See Figure 6 .

Using the graph in Figure 5 , ⓐ find g − 1 ( 1 ) , g − 1 ( 1 ) , and ⓑ estimate g − 1 ( 4 ) . g − 1 ( 4 ) .

Finding Inverses of Functions Represented by Formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example, y y as a function of x — x — we can often find the inverse function by solving to obtain x x as a function of y . y .

Given a function represented by a formula, find the inverse.

• Make sure f f is a one-to-one function.
• Solve for x . x .
• Interchange x x and y . y .
• Replace y y with f - 1 ( x ) f - 1 ( x ) . (Variables may be different in different cases, but the principle is the same.)

Inverting the Fahrenheit-to-Celsius Function

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

By solving in general, we have uncovered the inverse function. If

In this case, we introduced a function h h to represent the conversion because the input and output variables are descriptive, and writing C − 1 C − 1 could get confusing.

Solve for x x in terms of y y given y = 1 3 ( x − 5 ) . y = 1 3 ( x − 5 ) .

Solving to Find an Inverse Function

Find the inverse of the function f ( x ) = 2 x − 3 + 4. f ( x ) = 2 x − 3 + 4.

So f − 1 ( y ) = 2 y − 4 + 3 f − 1 ( y ) = 2 y − 4 + 3 or f − 1 ( x ) = 2 x − 4 + 3. f − 1 ( x ) = 2 x − 4 + 3.

The domain and range of f f exclude the values 3 and 4, respectively. f f and f − 1 f − 1 are equal at two points but are not the same function, as we can see by creating Table 5 .

Solving to Find an Inverse with Radicals

Find the inverse of the function f ( x ) = 2 + x − 4 . f ( x ) = 2 + x − 4 .

So f − 1 ( x ) = ( x − 2 ) 2 + 4. f − 1 ( x ) = ( x − 2 ) 2 + 4.

The domain of f f is [ 4 , ∞ ) . [ 4 , ∞ ) . Notice that the range of f f is [ 2 , ∞ ) , [ 2 , ∞ ) , so this means that the domain of the inverse function f − 1 f − 1 is also [ 2 , ∞ ) . [ 2 , ∞ ) .

The formula we found for f − 1 ( x ) f − 1 ( x ) looks like it would be valid for all real x . x . However, f − 1 f − 1 itself must have an inverse (namely, f f ) so we have to restrict the domain of f − 1 f − 1 to [ 2 , ∞ ) [ 2 , ∞ ) in order to make f − 1 f − 1 a one-to-one function. This domain of f − 1 f − 1 is exactly the range of f . f .

What is the inverse of the function f ( x ) = 2 − x ? f ( x ) = 2 − x ? State the domains of both the function and the inverse function.

Finding Inverse Functions and Their Graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function f ( x ) = x 2 f ( x ) = x 2 restricted to the domain [ 0 , ∞ ) , [ 0 , ∞ ) , on which this function is one-to-one, and graph it as in Figure 7 .

Restricting the domain to [ 0 , ∞ ) [ 0 , ∞ ) makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic function is the square root function, that is, f − 1 ( x ) = x . f − 1 ( x ) = x . What happens if we graph both f f and f − 1 f − 1 on the same set of axes, using the x - x - axis for the input to both f and   f − 1 ? f and   f − 1 ?

We notice a distinct relationship: The graph of f − 1 ( x ) f − 1 ( x ) is the graph of f ( x ) f ( x ) reflected about the diagonal line y = x , y = x , which we will call the identity line, shown in Figure 8 .

This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

Finding the Inverse of a Function Using Reflection about the Identity Line

Given the graph of f ( x ) f ( x ) in Figure 9 , sketch a graph of f − 1 ( x ) . f − 1 ( x ) .

This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of ( 0 , ∞ ) ( 0 , ∞ ) and range of ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , so the inverse will have a domain of ( − ∞ , ∞ ) ( − ∞ , ∞ ) and range of ( 0 , ∞ ) . ( 0 , ∞ ) .

If we reflect this graph over the line y = x , y = x , the point ( 1 , 0 ) ( 1 , 0 ) reflects to ( 0 , 1 ) ( 0 , 1 ) and the point ( 4 , 2 ) ( 4 , 2 ) reflects to ( 2 , 4 ) . ( 2 , 4 ) . Sketching the inverse on the same axes as the original graph gives Figure 10 .

Draw graphs of the functions f f and f − 1 f − 1 from Example 8 .

Is there any function that is equal to its own inverse?

Yes. If f = f − 1 , f = f − 1 , then f ( f ( x ) ) = x , f ( f ( x ) ) = x , and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because

Any function f ( x ) = c − x , f ( x ) = c − x , where c c is a constant, is also equal to its own inverse.

Access these online resources for additional instruction and practice with inverse functions.

• Inverse Functions
• One-to-one Functions
• Inverse Function Values Using Graph
• Restricting the Domain and Finding the Inverse

3.7 Section Exercises

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Why do we restrict the domain of the function f ( x ) = x 2 f ( x ) = x 2 to find the function’s inverse?

Can a function be its own inverse? Explain.

Are one-to-one functions either always increasing or always decreasing? Why or why not?

How do you find the inverse of a function algebraically?

Show that the function f ( x ) = a − x f ( x ) = a − x is its own inverse for all real numbers a . a .

For the following exercises, find f − 1 ( x ) f − 1 ( x ) for each function.

f ( x ) = x + 3 f ( x ) = x + 3

f ( x ) = x + 5 f ( x ) = x + 5

f ( x ) = 2 − x f ( x ) = 2 − x

f ( x ) = 3 − x f ( x ) = 3 − x

f ( x ) = x x + 2 f ( x ) = x x + 2

f ( x ) = 2 x + 3 5 x + 4 f ( x ) = 2 x + 3 5 x + 4

For the following exercises, find a domain on which each function f f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f f restricted to that domain.

f ( x ) = ( x + 7 ) 2 f ( x ) = ( x + 7 ) 2

f ( x ) = ( x − 6 ) 2 f ( x ) = ( x − 6 ) 2

f ( x ) = x 2 − 5 f ( x ) = x 2 − 5

Given f ( x ) = x 2 + x f ( x ) = x 2 + x and g ( x ) = 2 x 1 − x : g ( x ) = 2 x 1 − x :

• ⓐ Find f ( g ( x ) ) f ( g ( x ) ) and g ( f ( x ) ) . g ( f ( x ) ) .
• ⓑ What does the answer tell us about the relationship between f ( x ) f ( x ) and g ( x ) ? g ( x ) ?

For the following exercises, use function composition to verify that f ( x ) f ( x ) and g ( x ) g ( x ) are inverse functions.

f ( x ) = x − 1 3 f ( x ) = x − 1 3 and g ( x ) = x 3 + 1 g ( x ) = x 3 + 1

f ( x ) = − 3 x + 5 f ( x ) = − 3 x + 5 and g ( x ) = x − 5 − 3 g ( x ) = x − 5 − 3

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f ( x ) = x f ( x ) = x

f ( x ) = 3 x + 1 3 f ( x ) = 3 x + 1 3

f ( x ) = −5 x + 1 f ( x ) = −5 x + 1

f ( x ) = x 3 − 27 f ( x ) = x 3 − 27

For the following exercises, determine whether the graph represents a one-to-one function.

For the following exercises, use the graph of f f shown in Figure 11 .

Find f ( 0 ) . f ( 0 ) .

Solve f ( x ) = 0. f ( x ) = 0.

Find f − 1 ( 0 ) . f − 1 ( 0 ) .

Solve f − 1 ( x ) = 0. f − 1 ( x ) = 0.

For the following exercises, use the graph of the one-to-one function shown in Figure 12 .

Sketch the graph of f − 1 . f − 1 .

Find f ( 6 ) and  f − 1 ( 2 ) . f ( 6 ) and  f − 1 ( 2 ) .

If the complete graph of f f is shown, find the domain of f . f .

If the complete graph of f f is shown, find the range of f . f .

For the following exercises, evaluate or solve, assuming that the function f f is one-to-one.

If f ( 6 ) = 7 , f ( 6 ) = 7 , find f − 1 ( 7 ) . f − 1 ( 7 ) .

If f ( 3 ) = 2 , f ( 3 ) = 2 , find f − 1 ( 2 ) . f − 1 ( 2 ) .

If f − 1 ( − 4 ) = − 8 , f − 1 ( − 4 ) = − 8 , find f ( − 8 ) . f ( − 8 ) .

If f − 1 ( − 2 ) = − 1 , f − 1 ( − 2 ) = − 1 , find f ( − 1 ) . f ( − 1 ) .

For the following exercises, use the values listed in Table 6 to evaluate or solve.

Find f ( 1 ) . f ( 1 ) .

Solve f ( x ) = 3. f ( x ) = 3.

Solve f − 1 ( x ) = 7. f − 1 ( x ) = 7.

Use the tabular representation of f f in Table 7 to create a table for f − 1 ( x ) . f − 1 ( x ) .

For the following exercises, find the inverse function. Then, graph the function and its inverse.

f ( x ) = 3 x − 2 f ( x ) = 3 x − 2

f ( x ) = x 3 − 1 f ( x ) = x 3 − 1

Find the inverse function of f ( x ) = 1 x − 1 . f ( x ) = 1 x − 1 . Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

Real-World Applications

To convert from x x degrees Celsius to y y degrees Fahrenheit, we use the formula f ( x ) = 9 5 x + 32. f ( x ) = 9 5 x + 32. Find the inverse function, if it exists, and explain its meaning.

The circumference C C of a circle is a function of its radius given by C ( r ) = 2 π r . C ( r ) = 2 π r . Express the radius of a circle as a function of its circumference. Call this function r ( C ) . r ( C ) . Find r ( 36 π ) r ( 36 π ) and interpret its meaning.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, t , t , in hours given by d ( t ) = 50 t . d ( t ) = 50 t . Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t ( d ) . t ( d ) . Find t ( 180 ) t ( 180 ) and interpret its meaning.

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• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/3-7-inverse-functions

© Jan 9, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.