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Given a real-world situation that can be modeled by a linear function or a graph of a linear function, determine and represent a reasonable domain and range of the linear function by using inequalities.

TEKS Standards and Student Expectations

A(2)(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real‐world situations, both continuous and discrete; and represent domain and range using inequalities

Resource Objective(s)

Given a verbal statement or a graph of a linear function, determine its domain and range.

Essential Questions

What is domain?

What is range?

Which variable is the independent variable?

Which variable is the dependent variable?

Vocabulary

Determining the Domain and Range Modeled by a Linear Function

To determine the domain of a given situation, identify all possible x-values, or values of the independent variable. To determine the range of a given situation, identify all possible y-values, or values of the dependent variable.

Example 1
A clown at a birthday party can blow up five balloons per minute. The relationship between the number of balloons inflated and the time that has passed can be expressed with the equation y = 5x, where x is the number of minutes passed and y is the number of balloons inflated. Find the domain and range of the relations.

In this example, the independent variable (x) is the number of minutes. The possible x-values include all real numbers greater than or equal to 0, since time can be measured in fractional parts of a minute.

The dependent variable (y) is the number of balloons inflated. The possible y-values include all real numbers greater than or equal to 0.

Therefore, the domain is {x ≥ 0}, and the range is {y ≥ 0}.

DVD Rental Example

An online DVD rental site charges a monthly membership fee of $10, plus$4 per DVD that is rented. The relationship between the number of DVDs rented and the total charge per month can be expressed with the equation y = 4x + 10, where x is the number of DVDs rented and y is the total charge per month. Find the domain and range of this relationship.

In this example, the independent variable, x, is the number of DVDs rented. The possible x-values include all whole numbers, since only whole number of DVDs can be rented.

The dependent variable, y, is the total charge per month. The possible y-values include 10, 14, 18, 22, . . .

Therefore, the domain is {0,1,2,3, . . .}, and the range is {10,14,18, 22, . . .}.

Determining the Domain from a Graph

Identify the set of all the x-coordinates in a function’s graph to determine the domain.

In this example, the domain is {≥ 0}, since 0 is the lowest x-value and the arrow indicates the line continues to the right. The boundary number of 0 is included, since the dot is solid.

Identify the set of all the y-coordinates in the function’s graph to determine the range.

In this example, the range is {≥ -2}, since -2 is the lowest y-value and the arrow indicates the line continues upward. The boundary number of -2 is included, since the dot is solid.

Activity 1: Graphit Domain and Range

Click on the following link to go to the interactive Graphit page.

• Enter the following functions into the y(x) box.  Click "Plot/Update" and view the resulting graphs.
• Record the domain and range for each function in your OnTRACK Algebra Journal.
 Function Domain Range The cost to park in a garage is a $5 entry fee plus$2 per hour. y(x) = 2x + 5 The amount of snow that fell after midnight is 3 feet per hour. y(x) = 3x A swimming pool is draining at a rate of 1.5 feet per minute. y(x) = -1.5x