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.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf