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# 3.6: The Domain and Range of a Linear Function

Difficulty Level: At Grade Created by: CK-12

Introduction

In this lesson you will learn about the domain and the range of a relation. A relation is simply any set of points on the Cartesian plane. The domain of a relation is the set of possible values that ‘\begin{align*}x\end{align*}’ may have. The range of a relation is the set of possible values that ‘\begin{align*}y\end{align*}’ may have.

You will learn how to write the domain and range of a relation using interval notation. The domain and range of a relation is also described with respect to the number system to which it belongs. In Chapter One you learned

\begin{align*}N\end{align*} (natural numbers) = {1,2,3,4, …};

\begin{align*}W\end{align*} (whole numbers) = {0,1,2,3, … };

\begin{align*}I\end{align*} (integers) = {-3,-2,-1,0,1,2,3, …};

\begin{align*}Q\end{align*} (rational numbers) = {all numbers in the form \begin{align*}\frac{a}{b}\end{align*} where \begin{align*}b \ne 0\end{align*}, all terminating and repeating decimals};

\begin{align*}\overline{Q}\end{align*} (irrational numbers) = {all non-terminating, non-repeating decimals};

\begin{align*}R\end{align*} (real numbers) = {all rational and irrational numbers}. These number systems are very important when the domain and range of a relation are described using interval notation.

In addition, you will also learn how to write a suitable domain and range for a given problem.

## The Domain and the Range

Objectives

The lesson objectives for Domain and Range of a Linear Function are:

• Understanding the domain and expressing it using interval notation
• Understanding the range and expressing it using interval notation
• Determining the domain and range of a function from a graph
• Writing a suitable domain and range for a given problem.

Introduction

State the domain and the range of the following relation:

The points indicated on the graph are \begin{align*}\{(-5,-4),(-5,1),(-2,3),(2,1),(2,-4)\}\end{align*}

The domain is \begin{align*}\{-5,-2,2 \}\end{align*} and the range is \begin{align*}\{ -4,1,3\}\end{align*}.

Watch This

Guidance

Another way to describe a relation is by the set of data points. A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots. The example above in the concept content represents a discrete relation. A relation is said to be continuous if there are an infinite number of data points. The graph of a continuous relation is represented by lines.

The relation is a straight line that that begins at the point (2, 1). The straight line indicates that the relation is continuous. The straight line also indicates that all numbers are included in the domain and in the range. The domain and the range can be written in interval notation.

Another name for interval notation is set notation.

Example A

Which relations are discrete? Which relations are continuous? What is the domain? What is the range?

(i)

The graph appears as dots. Therefore, the relation is discrete. The domain is \begin{align*}\{1,2,4 \}\end{align*}. The range is \begin{align*}\{1,2,3,5 \}\end{align*}

(ii)

The graph appears as a straight line. Therefore, the relation is continuous.

\begin{align*}D=\{x|x \ \varepsilon \ R \} \quad R=\{y|y \ \varepsilon \ R \}\end{align*}

The graph appears as dots. Therefore, the relation is discrete.

The domain is \begin{align*}\{-1,0,1,2,3,4,5\}\end{align*}.

The range is \begin{align*}\{-2,-1,0,1,2,3,4\}\end{align*}

The graph appears as a straight line. Therefore, the relation is continuous.

\begin{align*}D=\{x|x \ \varepsilon \ R \} \quad R=\{y|y \ge -3, y \ \varepsilon \ R \}\end{align*}

Example B

Whether a relation is discrete or continuous can often be determined without a graph. As well, the domain and range can also be determined. Let’s examine the following toothpick pattern.

Complete the table below to determine the number of toothpicks needed for the pattern.

Pattern number \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 200
Number of toothpicks \begin{align*}(t)\end{align*}

Is the data continuous or discrete? Why?

What is the domain?

What is the range?

Pattern number \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 200
Number of toothpicks \begin{align*}(t)\end{align*} 7 12 17 22 27 \begin{align*}5n+2\end{align*} 1002

The number of toothpicks in any pattern number is the result of multiplying the pattern number by 5 and adding 2 to the product.

The number of toothpicks in pattern number 200 is:

\begin{align*}t&=5n+2\\ t&=5({\color{red}200})+2\\ t&=1000+2\\ t&=1002\end{align*}

The data must be discrete. The graph would be dots representing the pattern number and the number of toothpicks. It is impossible to have a pattern number or a number of toothpicks that are not natural numbers. Therefore, the points would not be joined.

The domain and range are:

\begin{align*}D=\{x|x \ \varepsilon \ N\} \quad R=\{y|y=5x+2, x \ \varepsilon \ N\} \ \ OR \ \ R=\{y|y \ge 7, y \ \varepsilon \ N\}\end{align*}

If the range is written in terms of a function, then the number system to which ‘\begin{align*}x\end{align*}’ belongs must be designated in the range.

Example C

Joseph drove from his summer home to his place of work. To avoid the road construction, Joseph decoded to travel the gravel road. After driving for 20 minutes he was 62 miles away from work and after driving for 40 minutes he was 52 miles away from work. Represent the problem on a graph and write a suitable domain and range for the situation.

To represent the problem on a graph, plot the points (20, 62) and (40, 52). The points can be joined with a straight line since the data is continuous. The distance traveled changes continuously as the time driving changes. The \begin{align*}y-\end{align*}intercept represents the distance from Joseph’s summer home to his place of work. This distance is approximately 72 miles. The \begin{align*}x-\end{align*}intercept represents the time it took Joseph to drive from his summer home to work. This time is approximately 145 minutes. In the next chapter you will learn how to determine the exact values of these points from the graph.

Time cannot be a negative quantity. Therefore, the smallest value for the number of minutes would have to be zero. This represents the time before Joseph began his trip. A suitable domain for this problem is \begin{align*}D=\{x|0 \le x \le 145, x \ \varepsilon \ R\}\end{align*}

The distance from his summer home to work cannot be a negative quantity. This distance is represented on the \begin{align*}y-\end{align*}axis as the \begin{align*}y-\end{align*}intercept and is the distance before he begins to drive. A suitable range for the problem is \begin{align*}R=\{y|0 \le y \le 72, y \ \varepsilon \ R\}\end{align*}

The domain and range often depend upon the quantities presented in the problem. In the above problem, the quantities of time and distance could not be negative. As a result, the values of the domain and the range had to be positive.

Vocabulary

Continuous Data
A relation is said to be continuous if there are an infinite number of data points. The graph of a continuous relation is represented by lines.
Discrete Data
A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots.
Domain
The domain of a relation is the set of possible values that ‘\begin{align*}x\end{align*}’ may have.
Range
The range of a relation is the set of possible values that ‘\begin{align*}y\end{align*}’ may have.
Coordinates
The coordinates are the ordered pair \begin{align*}(x, y)\end{align*} that represents a point on the Cartesian plane.

Guided Practice

1. Which relations are discrete? Which relations are continuous?

(i)

(ii)

2. State the domain and the range for each of the following relations:

(i)

(ii)

3. A computer salesman’s wage consists of a monthly salary of $200 plus a bonus of$100 for each computer sold.

(a) Complete the following table of values:

Number of computers sold 0 2 5 10 18
Wages in dollars for the month ($) (b) Sketch the graph to represent the monthly salary ($), against the number \begin{align*}(N)\end{align*}, of computers sold.

(c) Use the graph to write a suitable domain and range for the problem.

1. (i) The graph clearly shows that the points are joined. Therefore the data is continuous.

(ii) The graph shows the plotted points as dots that are not joined. Therefore the data is discrete.

2. (i) The domain represents the values of ‘\begin{align*}x\end{align*}’. \begin{align*}D=\{x|-3\le x\le 3, x \ \varepsilon \ R\}\end{align*}

The range represents the values of ‘\begin{align*}y\end{align*}’. \begin{align*}R=\{y|-3 \le y \le 3, y \ \varepsilon \ R \}\end{align*}

(ii) \begin{align*}D=\{x|x \ \varepsilon \ R\}\end{align*}

\begin{align*}R=\{y|-4 \le y \le 4, y \ \varepsilon \ R\}\end{align*}

3.

Number of computers sold 0 2 5 10 18
Wages in dollars for the month ($)$200 $400$700 $1200$2000

(c) The graph shows that the data is continuous. The number of computers sold and must be whole numbers. The wages must be natural numbers.

A suitable domain is \begin{align*}D=\{x|x \ge 0, x \ \varepsilon \ W\}\end{align*}

A suitable domain is \begin{align*}R=\{y|y \ge 200, y \ \varepsilon \ N\}\end{align*}

Summary

In this lesson you have learned that data that is represented by points that are plotted on a Cartesian grid but are not joined are referred to as discrete. Data that is represented by points that are plotted and joined on the Cartesian grid are referred to as continuous data. Data, whether it is discrete or continuous, also has values that are applicable to both the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes. The values that apply to the independent variable \begin{align*}(x)\end{align*} are called the domain of the function. The values that apply to the dependent variable \begin{align*}(y)\end{align*} are called the range of the function. The domain and the range are determined from the graph or from the word problem that is represented on the graph. These values must be suitable for both phenomena.

Problem Set

For each of the following, state whether the data is discrete or continuous. Also, write the domain and the range for each graph.

(a)

(b)

(c)

(d)

Examine the following patterns. Complete the table below each pattern. Write a suitable domain and range for each pattern and tell if the data is discrete or continuous.

(a)

Number of Cubes \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 200
Number of visible faces \begin{align*}(f)\end{align*}

(b)

Number of triangles \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 100
Number of toothpicks \begin{align*}(t)\end{align*}

(c)

Pattern Number \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ...' 100
Number of dots \begin{align*}(d)\end{align*}

1. (a) The points are joined. The data is continuous. The domain is: \begin{align*}D=\{x|-2 \le x \le 2, x \ \varepsilon \ R\}\end{align*} The range is: \begin{align*}R=\{y|-2 \le y \le 2, y \ \varepsilon \ R\}\end{align*} (b) All the lines represent continuous data. Line 1 The domain is: \begin{align*}D=\{x| x \ \varepsilon \ R\}\end{align*} The range is: \begin{align*}R=\{y| y \ \varepsilon \ R\}\end{align*} Line 2 The domain is: \begin{align*}D=\{x|0 \le x \le 3, x \ \varepsilon \ R\}\end{align*} The range is: \begin{align*}R=\{y|-4 \le y < 5, y \ \varepsilon \ R\}\end{align*} Line 3 The domain is: \begin{align*}D=\{x|-1 \le x \le 1, x \ \varepsilon \ R\}\end{align*} The range is: \begin{align*}R=\{y|-5 \le y \le 3, y \ \varepsilon \ R\}\end{align*} (c) The points are not joined. The data is discrete. The domain is: \begin{align*}D=\{-3,-1,1,2,3\}\end{align*} The range is: \begin{align*}R=\{-4,-3,1,2,4,5\}\end{align*} (d) The points are joined. The data is continuous. The domain is: \begin{align*}D=\{x|-2 \le x \le 3, x \ \varepsilon \ R\}\end{align*} The range is: \begin{align*}D=\{y|-3 \le y \le 1, y \ \varepsilon \ I\}\end{align*}
2. (a)
Number of Cubes \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 200
Number of visible faces \begin{align*}(f)\end{align*} 6 10 14 18 22 \begin{align*}4n+2\end{align*} 802

The data is discrete.

The domain is: \begin{align*}D=\{x|x \ \varepsilon \ N\}\end{align*}

The range is: \begin{align*}R=\{y|y=4x+2, x \ \varepsilon \ N\}\end{align*}

(b)

Number of triangles \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 100
Number of toothpicks \begin{align*}(t)\end{align*} 3 5 7 9 11 \begin{align*}2n+1\end{align*} 201

The data is discrete.

The domain is: \begin{align*}D=\{x|x \ \varepsilon \ N\}\end{align*}

The range is: \begin{align*}R=\{y|y=2x+1, x \ \varepsilon \ N\}\end{align*}

(c)

Pattern Number \begin{align*}(n)\end{align*} 1 2 3 4 5 ... \begin{align*}n\end{align*} ... 100
Number of dots \begin{align*}(d)\end{align*} 4 9 16 25 36 \begin{align*}(n+1)^2\end{align*} \begin{align*}(101)^2\end{align*}

The data is discrete.

The domain is: \begin{align*}D=\{x|x \ \varepsilon \ N\}\end{align*}

The range is: \begin{align*}R=\{y|y=(x+1)^2, x \ \varepsilon \ N\}\end{align*}

## Summary

In this lesson you have learned that data that is represented by points that are plotted on a Cartesian grid but are not joined are referred to as discrete. Data that is represented by points that are plotted and joined on the Cartesian grid are referred to as continuous data. Data, whether it is discrete or continuous, also has values that are applicable to both the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes. The values that apply to the independent variable \begin{align*}(x)\end{align*} are called the domain of the function. The values that apply to the dependent variable \begin{align*}(y)\end{align*} are called the range of the function. The domain and the range are determined from the graph or from the word problem that is represented on the graph. These values of the domain and the range must be suitable for the graph and for the word problem. It is also important to identify the number system to which the domain and the range belong.

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