# Let's Get Started

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* = 5*x*, 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* = 4*x* + 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 {*x *≥ 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 {*y *≥ -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.

# Another Example

# 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*) = 2*x*+ 5The amount of snow that fell after midnight is 3 feet per hour. y( *x*) = 3*x*A swimming pool is draining at a rate of 1.5 feet per minute. y( *x*) = -1.5*x*