**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