Lesson 5: Combining Functions

What does it mean to combine two functions?

How can functions be added, subtracted, multiplied, and divided?

How do we write combined functions using function notation?

How do we determine the domain of a combined function?

What restrictions must be considered when dividing functions?

How can combining functions help us model more complex relationships?

Combined functions

Sum of functions

Difference of functions

Product of functions

Quotient of functions

Function notation

Domain

Restriction

Undefined value

HSA-APR.A.1

Understand that polynomials form a system analogous to integers, meaning they are closed under addition, subtraction, and multiplication.

HSF-BF.A.1

Write a function that describes a relationship between two quantities.

HSF-BF.B.3

Identify the effect on the graph of replacing one function with combinations of functions.

Lesson Description

Students learn how to combine two functions algebraically by performing operations such as addition, subtraction, multiplication, and division.

Students will:

Use function notation to express operations on functions

Simplify combined function expressions

Determine the domain of the resulting function, including identifying restrictions caused by division

Students will also interpret how combining functions can represent multiple processes occurring simultaneously.

Purpose

The purpose of this lesson is to help students understand that functions can interact with each other algebraically, just like numbers.

This concept builds a foundation for later topics including:

Composite functions

Function transformations

Modeling complex systems

Advanced algebra and calculus concepts

Understanding combined functions also helps students interpret more complex real-world models where multiple relationships influence a single outcome.

Primary Level: DOK 2 – Skills and Concepts

Support for Developing Learners

Review basic operations with algebraic expressions.

Provide structured step-by-step examples.

Use color-coding to show which parts belong to each function.

Begin with simple linear functions before progressing to polynomials or rational functions.

On-Level Strategies

Students practice combining functions using symbolic notation.

Partner work solving and checking combined function problems.

Domain identification exercises using tables and graphs.

Enrichment / Extension

Introduce real-world scenarios involving combined processes.

Challenge students to determine combined functions that produce specific outputs.

Explore how combined functions behave graphically using graphing technology.

exit slips and quiz

textbook

• Desmos Graphing Calculator

• TI-83 / TI-84 Graphing Calculators

• SmartBoard demonstrations of combined functions