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A2-F-2-2

Page history last edited by Brenda Butz 6 years, 3 months ago

A2.F.2.2 Combine functions by composition and recognize that g(x)=f^-1(x), the inverse function of f(x), if and only if f(g(x))=g(f(x))=x.

In a Nutshell

The composition of functions is a powerful process, by which, a new function is produced, and the domain and range must be explored.

Student Actions	Teacher Actions
Students develop mathematical reasoning as they determine which procedures/strategies are efficient for combining functions by composition. Students will demonstrate a deep conceptual understanding as they analyze functions and inverse functions. Students will develop a productive mathematical disposition as they make a connection between the algebraic procedure of proving two functions are inverses to the graphs of the two functions. Students will develop the ability to make conjectures, model and generalize about the graph the function and its inverse and use the line y=x and the definition of inverse functions to justify their answer. Students will demonstrate a deep conceptual understanding of the composition of two functions, algebraically, numerically and graphically.	Implement tasks that engage students and promote mathematical reasoning of combining functions by composition through various methods including algebraic, numeric, and graphic methods Pose purposeful questions to ensure conceptual understanding of the definition of inverse functions and how to prove two functions are inverses. Elicit and use evidence of student thinking by providing students with various representations of functions and their inverses (algebraic, numeric, graphic) and ask students to identify and justify the function and its inverse Facilitate mathematical discussion about how the domain and range is affected in the composition of two functions. http://www.mhhe.com/math/precalc/barnettpc5/graphics/barnett05pcfg/ch02/others/bpc5_ch02-05.pdf Support productive struggle in learning mathematics by providing students with f(g(x))=h(x) and ask to determine f(x) and g(x) (Decomposition). ( ie. f(g(x)) = h(x) and h(x) = , determine f(x) and g(x).) Give students a verbal description of a function and ask them to work together to write a verbal description of the inverse of a function.
Key Understandings	Misconceptions
Procedural: Use the definition of an inverse function to verify that two functions are inverses. Conceptual: Combine functions through composition algebraically, numerically, and graphically. Verify that functions are inverses graphically, numerically, algebraically, and verbally.	Procedural: When performing function composition, students do not know which function to perform first. Conceptual: Students think that (f(g(x)) means to multiply f, g, and x. Students have difficulty understanding there is a difference between f(g(x)) and (fg)(x). Students are not able to apply the formal definition of inverse functions through composition.