# ALEX Lesson Plan

## Building Functions:  Composition of Functions

You may save this lesson plan to your hard drive as an html file by selecting "File", then "Save As" from your browser's pull down menu. The file name extension must be .html.

This lesson provided by:
 Author: Lorie White System: Lauderdale County School: Rogers High School
General Lesson Information
 Lesson Plan ID: 35604 Title: Building Functions:  Composition of Functions Overview/Annotation: This lesson provides a review of evaluating functions and finding function rules as well as an introduction to the composition of functions. The review is accomplished through the use of an online exploration using a function machine. The idea of a function machine is also used to explain the composition of functions. Instruction is provided in finding the composition using several different representations of functions (input/output tables, graphs, and function rules). This lesson results from the ALEX Resource Gap Project.
Associated Standards and Objectives
Content Standard(s):
Mathematics
MA2019 (2019)
Algebra I with Probability
17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function Input Output
G Amount of studying: s Grade in course: G(s)
S Grade in course: g Amount of screen time: S(g)
T Amount of screen time: t Number of follers: T(t)
Unpacked Content
Evidence Of Student Attainment:
Students:
Given different types of standard functions
• Use arithmetic operations to combine functions in context.
• Use function composition to combine functions in context.
• Write, evaluate, and interpret combined functions in context.
Teacher Vocabulary:
• Function composition
Knowledge:
Students know:
• Techniques to combine functions using arithmetic operations.
• Techniques for combining functions using function composition.
Skills:
Students are able to:
• Accurately develop a model that shows the functional relationship between two quantities.
• Accurately create a new function through arithmetic operations of other functions.
• Present an argument to show how the function models the relationship between the quantities.
Understanding:
Students understand that:
• Arithmetic combinations of functions may be used to improve the fit of a model.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Prior Knowledge Skills:
• Explain the distributive property.
• Give examples of the properties of operations including distributive.
• Combine like terms of a given expression.
• Recognize the correct order to solve expressions with more than one operation.
• Calculate a numerical expression (Ex. V=(4x4x4).
• Choose the correct value to replace each variable in the algebraic expression (Substitution).
• Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).

Local/National Standards:

Primary Learning Objective(s):

• Students will be able to write a function rule given domain and range values.
• Students will be able to find the range values of a composition of two functions given the two function rules and a domain value.
• Students will be able to write a function rule for the composition of two functions given the function rules for the two functions.
• Students will be able to find the value of the composition of two functions given the graphs of the two functions.