Real-World Domain and Range

4.13.4 Real-World Domain and Range

4.13.4 • Real-World Domain and Range

Activity

A discrete function has distinct separate values in the domain and range, rather than an interval of values.

1. Below is a graph from the previous lesson that graphs the number of students attending a camp and the amount, in dollars, the camp earns. There must be 5 students to hold the camp.

A scatter plot showing a positive linear relationship between number of students (x-axis, 0–20) and amount in dollars (y-axis, 0–700). As the number of students increases, the amount in dollars rises.

a. What are the $x$ -values of this function that make up the domain?

Compare your answers:

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

b. Why can’t we say the domain is all values from 5 to 16?

Compare your answers:

The number of students cannot be fractions or decimals, so those values are not in the domain.

c. What are the $y$ -values of this function that make up the range?

Compare your answers:

200, 250, 300, 350, 400, 450, 500, 550, 600, 650

A continuous function refers to a function that does not have any breaks in its graph. The domain and range of a continuous function are often represented by an interval that can be written as an inequality.

2. The graph below represents a continuous function. The function, $P$ , is the cost of parking in a garage for $t$ hours. It is represented by the function $P (t) = 5 h + 10$ .

$A graphed line with a $y$-intercepts at (0, 10), passing through the points (2, 20) and (4, 30).$

a. What does $P (0)$ represent?

Compare your answer:

It represents the starting price of parking for the day.

b. In words, what are the values of the domain?

Compare your answer:

The domain includes the values where $x$ is at least 0.

c. How can you write the domain values as an inequality?

Compare your answer:

$x \geq 0$

d. In words, what are the values of the range?

Compare your answer:

The range includes the values where $y$ is at least 10.

e. How can you write the range values as an inequality?

Compare your answer:

$y \geq 10$

Self Check

Which of the following correctly describes the domain and range of the graph below?

GRAPH OF A LINE FOR ALL X GREATER THAN OR EQUAL TO 2, WITH POINT SHOWN AT (2, 3), AND PASSING THROUGH THE POINTS (5, 9) AND (8, 15).

Domain: $x > 2$
Range: $y > 3$
Domain: $x\leq2$
Range: $y\leq3$
Domain: $x\geq2$
Range: $y\geq3$
Domain: $x\geq3$
Range: $y\geq2$

Additional Resources

Domain and Range as Inequalities

Looking at the linear function below, use inequalities to represent the domain and range.

Graph of a line for all x greater than or equal to 5, with point shown at (5, 0), and passing through the points (10, 5), and (15, 10).

Domain:

Since the domain is all $x$ -values, look left to right for the least to greatest values.

The smallest $x$ -value on this linear function is 5, and then the line continues to the right forever.

So, the domain is all values that are at least 5.

As an inequality, write this as $x \geq 5$ .

Range:

Since the range is all $y$ -values, look from the bottom to the top for the least to greatest values.

The smallest $y$ -value is 0, and then the line continues up forever.

So, the range is all values that are at least 0.

An an inequality, the range is $y \geq 0$ .

Try it

Domain and Range as Inequalities

Looking at the linear function below, use inequalities to represent the domain and range.

Graph of a line for all x greater than or equal to 5, with point shown at (5, 6), and passing through the points (7, 10) and (10, 16).

Your answers may vary, but here are some examples:

Here is how to write the domain and range of the linear function as an inequality:

Domain:

Since the domain is all $x$ -values, look left to right for the least to greatest values.

The smallest $x$ -value on this linear function is 5, and then the line continues to the right forever.

So, the domain is all values that are at least 5.

As an inequality, write this as $x \geq 5$ .

Range:

Since the range is all $y$ -values, look from the bottom to the top for the least to greatest values.

The smallest $y$ -value is 6, and then the line continues up forever.

So, the range is all values that are at least 6.

An an inequality, the range is $y \geq 6$ .