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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 02: Composition and Inverses of Functions SUGGESTED DURATION : 11 days

#### Unit Overview

Introduction
This unit bundles student expectations that address composition of two or more functions and the inverse of a function using multiple representations. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards, including application, tools and techniques, communication, representations, relationships, and justifications, should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Algebra II Units 01, 06, 07, and 10, students described and analyzed the relationship between a function and its inverse, including restrictions on the domain where appropriate. Students used the composition of two functions to determine if the functions were inverses.

During this Unit
Students determine and represent an inverse function, when it exists, for a given function over the original domain or a subset of its domain using graphical, tabular, and algebraic representations. If necessary, students restrict the domain of the original function in order to maintain functionality of the inverse. Students use composition of functions to determine if two functions are inverses. Students write compositions of functions as f(g(x)) or (f  g)(x). Students represent a given function as a composite of two or more functions using numeric, tabular, graphical, and algebraic methods. Students demonstrate that function composition is not always commutative using various representations. Students use the composition of two functions to model and solve real-world problems.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Precalculus

After this Unit
In Units 03 – 05 and 08, students will continue to apply function composition and inverses to polynomial, power, rational, exponential, logarithmic, and trigonometric functions in mathematical and real-world problem situations. In subsequent mathematics courses, students will continue to apply these concepts as they arise in problem situations.

Function analysis serves as the foundation for college readiness. Analyzing, representing, and modeling with functions are emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VI. Functions B2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to a 2007 report published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the TxCCRS, which calls for students to be able to understand and analyze features of a function to model real world situations. Algebraic models allow us to efficiently visualize and analyze the vast amount of interconnected information that is contained in a functional relationship; these tools are particularly helpful as the mathematical models become increasingly complex (National Research Council, 2005). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (NRC, 2005, p. 353). Lastly, students need to be involved in metacognitive engagement in mathematics as they problem solve and reflect on their solutions and strategies; this is particularly important as students transition into more abstract mathematics, where fewer “clues” may exist warning students of a mathematical misstep (NRC, 2005). An important mathematical technique is to decompose a situation, analyze its pieces, and then recompose the pieces back together in order to draw conclusions; this technique can be used with functions in order to see the relationships that exist between combined, composed, and transformed functions (Cooney, Beckmann, & Lloyd, 2010). These skills are further applied in Calculus, where functions are decomposed in order to make use of derivative and integration rules for sums, differences, products, quotients, and compositions of functions.

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Research Council. (2005). How Students Learn: Mathematics in the Classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationship?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• What relationships exist between a function and its inverse?
• How are the key attributes of a function related to the key attributes of its inverse?
• How can the inverse of a function be determined and represented?
• How can function composition be used to analyze relationships between functions?
• How can function composition be used to model and analyze problem situations?
• Functions can be represented in various ways (including algebraically, graphically, verbally, and numerically) with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions, Equations, and Inequalities
• Functions
• Inverse
• Compositions
• Patterns, Operations, and Properties
• Relations and Generalizations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationship?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How can function composition be used to analyze relationships between functions?
• Why is the composition of functions not always commutative?
• How can function composition be used to model and analyze problem situations?
• Functions can be represented in various ways (including algebraically, graphically, verbally, and numerically) with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions, Equations, and Inequalities
• Functions
• Compositions
• Patterns, Operations, and Properties
• Relations and Generalizations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may confuse the notation for function composition, (f  g)(x), and the notation for the product of two functions, (fg)(x).
• Some students may think that function composition is commutative for all functions f and g.
• Some students may think that any two functions can be composed. Consider the case of f(x) = –x2 and g(x) = log(x). The composition g(f(x)) = log(–x2) is nonsensical, since there are no x’s in the domain of f such that f(x) is in the domain of g(x).
• Some students may think that the inverse of a function, f–1(x), is a reflection of the function across the x-axis (i.e. f–1(x) = – f(x)).
• Some students may forget to restrict the domain of the original function f(x) so that f –1(x) is truly its inverse. For example, f–1(x) = is the inverse of when the domain of f(x) is restricted to x ≥ 0.
• Some students may confuse the notation for inverse function, f–1(x), with the notation for a reciprocal, x–1. For example, students might incorrectly believe that given f(x) = , f–1(x) = instead of the correct answer, f–1(x) = x2 – 2.

#### Unit Vocabulary

• Commutative property – mathematical property in which the final result of an operation is not changed when the order of the operands is switched
• Composition of functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other function
• Inverse of a function – function that undoes the original function. When composed f(f–1(x)) = x and f–1(f(x)) = x.

Related Vocabulary:

 Composite function Decompose Domain Function notation Functionality Inverse function Ordered pairs Range Restriction Subset Transformations
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
P.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
P.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
P.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII. A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2A Use the composition of two functions to model and solve real-world problems.

Use

THE COMPOSITION OF TWO FUNCTIONS

Including, but not limited to:

• Composition of functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other function
• Notation for the composition of two functions
• g(f(x))
• (g ○ f)(x)
• Methods for the composition of two functions
• Numeric
• Tabular
• Graphical
• Algebraic
• Compositions with absolute value functions

To Model, To Solve

THE COMPOSITION OF TWO FUNCTIONS IN REAL-WORLD PROBLEMS

Including, but not limited to:

• Models of the composition of two functions
• Verbal
• Algebraic
• Tabular
• Graphical
• Domain and range restrictions to real-world problem situations
• Solution to composition of two functions
• Algebraically by substituting one function into the other function
• Tabularly by using the range of one function as the domain of the other function
• Graphically by using the range of one function as the domain of the other function
• Domain and range restrictions to real-world problem situations

Note(s):

• Algebra II used composition of functions to identify inverse functions.
• Precalculus uses composition of functions to model real-world situations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
P.2B Demonstrate that function composition is not always commutative.

Demonstrate

THAT FUNCTION COMPOSITION IS NOT ALWAYS COMMUTATIVE

Including, but not limited to:

• Composition of functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other function
• Commutative property – mathematical property in which the final result of an operation is not changed when the order of the operands is switched
• Models to determine if the composition of two functions is commutative
• Numerically
• Symbolically
• Tabularly
• Graphically
• Verbally
• Method for determining commutativity of function composition
• Given two functions, f(x) and g(x)
• Evaluate g(f(x)).
• Evaluate f(g(x)).
• If g(f(x)) = f(g(x)) for all x, then the function composition is commutative.
• If g(f(x)) ≠ f(g(x)) for any x, then the function composition is not commutative.

Note(s):

• Algebra II used composition of functions to identify inverse functions.
• Precalculus uses composition of functions to determine commutativity.
• Various mathematical process standards will be applied to this student expectation as appropriate.
P.2C Represent a given function as a composite function of two or more functions.

Represent

A GIVEN FUNCTION AS A COMPOSITE OF TWO OR MORE FUNCTIONS

Including, but not limited to:

• Models to determine if the composition of two functions is commutative
• Numerically
• Symbolically
• Tabularly
• Graphically
• Verbally
• Decompose a given function to determine two or more functions that when composed create the given function.
• A sequence of operations to create a given function
• A sequence of transformations to create a given function

Note(s):

• Algebra II used composition of functions to identify inverse functions.
• Precalculus uses composition of functions to represent a function as a composite of two or more functions.
• Calculus will investigate the composition rule of derivatives.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
P.2E Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

Determine

AN INVERSE FUNCTION, WHEN IT EXISTS, FOR A GIVEN FUNCTION OVER ITS DOMAIN OR A SUBSET OF ITS DOMAIN

Represent

THE INVERSE OF A FUNCTION USING MULTIPLE REPRESENTATIONS

Including, but not limited to:

• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.
• Characteristics of inverse functions
• Domain of the function becomes an appropriate range of the inverse function.
• Range of the function becomes an appropriate domain of the inverse function.
• Composed as f(f –1(x)) = x and f –1(f(x)) = x
• Multiple representations
• Inverse function notation
• When a function f(x) has an inverse that is also a function, the inverse can be written with f –1(x).
• For the function f(x) = x + 4, the inverse function is f –1(x) = x – 4.
• For the function g(x) = x2:
• If the restricted domain of g(x) is x ≥ 0, then the inverse function is g1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g1(x) = –.
• Algebraic
• The inverse of a function can be found algebraically by:
• Writing the original function in “y = ” form
• Interchanging the x and y variables
• Solving for y
• A function’s inverse can be confirmed algebraically if both of the following are true: f(f –1(x)) = x and f –1(f(x)) = x.
• Tabular
• From the table of values for a given function, the tabular values of the inverse function can be found by switching the x- and y-values of each ordered pair.
• Graphical
• The graphs of a function and its inverse are reflections over the line y = x.
• Verbal description of the relationships between the domain and range of a function and its inverse
• Restrictions on the domain of the original function to maintain functionality
• Inverse functions over a subset of the domain of the original function

Note(s):