Chapter 3 (PDM)

Lesson Plan 3-2: Composition and Inverses of Functions

Content Standards and Benchmarks:

I,2,3: Expand their understanding of functions to include non-linear functions, composition of functions, inverses of functions, and piecewise and recursively defined functions.

IV,2,2: Develop an understanding of more complex representations of numbers, including exponential and logarithmic expressions, and select an appropriate representation to facilitate problem solving.

Objective:

          After reading the lesson the night before, and hearing my short lecture regarding the composition and inverses of functions, students will be able to compose multiple functions and find the inverses of functions.

Anticipatory Set:

          Conversation with the class about the women’s basketball team winning their 7^th league game.  Transition to the homework and how no one really likes to work with logarithmic functions.  Transition into what the composition of functions are and how to find the inverse of a function. 

Input:

Ø     x has to be an element of the domain of f for f(x) to be defined

Ø     f(x) has to be an element of the domain of g for g(x) to be defined

Ø     The domain of g of f is the intersection of the domain of f and the domain of g

Ø     Function composition is not commutative.

Ex: Gumball machine

Ø     Multiplication and composition looks similar but are very different

Ø     We use function composition to solve equations

Ø     Inverse of a function: if

         ¨ f: xày and g: yàx, f and g are inverses.

         ¨     Continually increasing or decreasing over a  

              continuous domain

         ¨     Passes Horizontal Line Test and Vertical Line Test

Ø     Limit domains of functions to make their inverses 1-1 functions

Ø     Check for inverses f(g(x)) =g(f(x))

Modeling:

v    Answer student questions from the night before.

v    Tie in today’s objectives with the questions that are asked.

Guided Practice:

v    As the students get more comfortable with the content, I will ask them to lead me through the problems.

v    I may even ask student to come up to the board to do that other students are confused about problems.

Independent practice:

v      Students will be assigned lesson 3-3 to read and do the C.A.R. problems for the next day.

Closure:

v      “What did we discuss today?”

Ø              How to tell if a function is 1-1

Ø              How to tell if a function has an inverse

Ø              Limit domains to make a function or its 

         inverse 1-1

Ø              Function composition is not commutative