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# Domain and Range of Linear Functions

- Author:
- Paige Eslick

- Topic:
- Functions, Linear Functions

GeoGebra Worksheet 1: Domain and Range of Linear Functions
Lesson Objectives:
• Practice restricting domain and range of linear functions
• Understand how domain affects the range in a function
• Determine what domain and range makes sense for a situation
Common Core Standards:
• HSF-BF.A.1 Build a function that models a relationship between two quantities.
Write a function that describes a relationship between two quantities.
• HSF-IF.B.5 Interpret functions that arise in applications in terms of the context.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes
Goal: Using GeoGebra “Domain and Range of Linear Functions” with sliders allows students to gain a better understanding of domain and range by interacting with a linear function and adjusting its domain.

The domain and range of linear functions is all real numbers, or negative infinity to positive infinity. Sometimes, however, the domain and range of a linear function may be restricted based on the information it represents. Let’s explore both of these scenarios.
Directions:
Open “Domain and Range of Linear Functions” in the GeoGebra program.
Notice two linear functions are shown: The green function f(x) = x/2 + 4, and the blue function p(x) = x/2 + 2.
1. What are the domain and range of the linear function f(x)?
Give all answers for domain and range in interval notation (using brackets and parenthesis).
Domain ________________________
Range ________________________
Now let’s practice restricting domain and range of a linear function. Notice the blue p(x) function does not continue forever in either direction; instead, its domain and range are restricted by two slider values. Practice moving the sliders and see how the graph of p(x) changes.
2. Set the sliders on two new values. Determine the new domain and range.
Domain ________________________
Range ________________________
Circle one: The ( domain / range ) of a function depends on the ( domain / range ).
Adjust the sliders again and find the new domain and range.
Domain ________________________
Range ________________________
Now let’s consider how the relationship that a linear function represents can affect its domain and range.
Example:
Tyler mows lawns after school each week. He is paid $10 for every lawn he mows. Write a linear function to represent the money he earns, y, based on the number of lawns he mows, x.
The amount of money Tyler earns can be expressed using the linear function ______________.
Consider this a linear function without any restrictions. Give the domain and range.
Domain ________________________
Range ________________________
However, this is not realistic for the situation this function represents. We must look at how the relationship of mowing lawns and earning money affects the domain and range of the function representing this scenario.
Tyler has a busy schedule with homework and sports practices. He can mow at most 8 lawns each week.
Tyler cannot mow a negative number of lawns. Since x represents the number of lawns Tyler mows each week, x must be greater than or equal to zero. Consider how this starting value affects the range: If Tyler mows zero lawns in a given week he will earn no money, thus the starting value of both the domain and range of the function y = 10x is ______. Also, we are told Tyler cannot mow more than 8 lawns each week. This is a restriction on the domain. From above, we know the domain of a function affects its range.
So, if Tyler mows at most 8 lawns one week, what is the most money he can earn that week?
______________________
Based on these restrictions, state the domain and range of the linear function representing the amount of money Tyler earns.
Domain ________________________
Range ________________________