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

#### Unit Overview

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.

After this unit, in Precalculus 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): II. Algebraic Reasoning B1, C1, D1, D2; III. Geometric Reasoning C1; VI. Statistical Reasoning B2, C3; VII. Functions B1, B2, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Inverses and composition of functions create new functions.

• What relationships and characteristics exist between a function and its inverse?
• What is the purpose of composition of functions?
• Why would a function need to be decomposed into two or more functions?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Multiple Representations
• Solve

Functions

• Attributes of Functions
• Composite of Functions
• Inverse of Functions
• Linear Functions
• Non-linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Composites of functions can be determined from multiple representations and used to model and solve real-world problems.

• How can a composite function be constructed using the equations of two or more functions?
• How can a composite function be constructed using tables of coordinate points of two or more functions?
• How can a composite function be constructed using graphs of two or more functions?
• How can the composition of functions be used to solve real-world problems?
• How is a function decomposed into a composition of two or more functions?
• Under what domain and range conditions can functions be composed?
• How do the attributes of newly composed functions compare to the attributes of original functions?
• Why is the composition of functions not always commutative?

The inverse of a function can be determined from multiple representations.

• How can the inverse of a function be determined from the graph of the function?
• How can the inverse of a function be determined from a table of coordinate points of the function?
• How can the inverse of a function be determined from the equation of the function?
• How are a function and its inverse distinguished symbolically?
• How are function compositions related to inverse functions?
• How do the attributes of inverse functions compare to the attributes of original functions?

The domain and range of the inverse of a function may need to be restricted in order for the inverse to also be a function.

• When must the domain of an inverse function be restricted?
• How does the relationship between a function and its inverse, including the restriction(s) on the domain, affect the restriction(s) on its range?
 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.

Algebraic Reasoning

• Multiple Representations

Functions

• Attributes of Functions
• Composite of Functions
• Non-linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Transformations can be modeled through function composition.

• How can a transformed function be written as a composition of two or more functions?
• Is there a unique decomposition that can be used to represent a transformed function? Explain.
• How can a function be decomposed by analyzing the operations?
• How do the attributes of newly composed functions compare to the attributes of the original functions?

#### 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, (f  g)(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
• 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 Creator 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 (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
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:
• VIII. Problem Solving and Reasoning
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:
• IX. Communication and Representation
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:
• IX. Communication and Representation
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:
• X. Connections
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:
• IX. Communication and Representation
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:
• III Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VI Statistical Reasoning
• B2 – Select and apply appropriate visual representations of data.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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.
• TxCCRS:
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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:
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 g-1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g-1(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):

• Algebra II analyzed the relationship between functions and inverses, such as quadratic and square root, or logarithmic and exponential, including necessary restrictions on the domain.
• Precalculus extends the analysis of inverses to include other types of functions, such as trigonometric and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VI Statistical Reasoning
• B2 – Select and apply appropriate visual representations of data.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections