# 3.6: Domain and Range

Difficulty Level: Advanced Created by: CK-12
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Practice Domain and Range of a Function

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Joseph drove from his summer home to his place of work. To avoid the road construction, Joseph decided 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.

### Guidance

The domain of a relation is the set of possible values that ‘x\begin{align*}x\end{align*}’ may have. The range of a relation is the set of possible values that ‘y\begin{align*}y\end{align*}’ may have. You can write the domain and range of a relation using interval notation and with respect to the number system to which it belongs. Remember:

• Z(integers)={3,2,1,0,1,2,3,}\begin{align*}Z(\text{integers})=\{-3,-2,-1,0,1,2,3, \ldots \}\end{align*}
• R(real numbers)={all rational and irrational numbers}\begin{align*}R(\text{real numbers})=\{\text{all rational and irrational numbers} \}\end{align*}.

These number systems are very important when the domain and range of a relation are described using interval notation.

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. A relation is said to be continuous if its graph is an unbroken curve with no "holes" or "gaps." The graph of a continuous relation is represented by a line or a curve like the one below. Note that it is possible for a relation to be neither discrete nor continuous.

The relation is a straight line that that begins at the point (2, 1). The fact that the points on the line are connected indicates that the relation is continuous. The domain and the range can be written in interval notation, as shown below:

#### Example A

Which relations are discrete? Which relations are continuous? For each relation, find the domain and range.

(i)

(ii)

(iii)

(iv)

Solution:

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

(ii) The graph appears as a straight line. Therefore, the relation is continuous. D={x|x  R}R={y|y  R}\begin{align*}D=\{x|x \ \in \ R \} \quad R=\{y|y \ \in \ R \}\end{align*}

(iii) The graph appears as dots. Therefore, the relation is discrete. The domain is {1,0,1,2,3,4,5}\begin{align*}\{-1,0,1,2,3,4,5\}\end{align*}. The range is {2,1,0,1,2,3,4}\begin{align*}\{-2,-1,0,1,2,3,4\}\end{align*}

(iv) The graph appears as a curve. Therefore, the relation is continuous. D={x|x  R}R={y|y3,y  R}\begin{align*}D=\{x|x \ \in \ R \} \quad R=\{y|y \ge -3, y \ \in \ R \}\end{align*}

#### Example B

Whether a relation is discrete, continuous, or neither can often be determined without a graph. The domain and range can be determined without a graph as well. Examine the following toothpick pattern.

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

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

Is the data continuous or discrete? Why?

What is the domain?

What is the range?

Solution:

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

tttt=5n+2=5(200)+2=1000+2=1002\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:

D={x|x  N}R={y|y=5x+2,x  N}\begin{align*}D=\{x|x \ \in \ N\} \quad R=\{y|y=5x+2, x \ \in \ N\} \end{align*}

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

#### Example C

Can you state the domain and the range of the following relation?

Solution:

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

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

#### Concept Problem Revisited

Joseph drove from his summer home to his place of work. To avoid the road construction, Joseph decided 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 y\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 x\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.

Time cannot be a negative quantity. Therefore, the smallest value for the number of minutes would have to be zero. This represents the time Joseph began his trip. A suitable domain for this problem is D={x|0x145,x  R}\begin{align*}D=\{x|0 \le x \le 145, x \ \in \ R\}\end{align*}

The distance from his summer home to work cannot be a negative quantity. This distance is represented on the y\begin{align*}y\end{align*}-axis as the y\begin{align*}y\end{align*}-intercept and is the distance before he begins to drive. A suitable range for the problem is R={y|0y72,y  R}\begin{align*}R=\{y|0 \le y \le 72, y \ \in \ R\}\end{align*}

The domain and range often depend on 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
A relation is said to be continuous if it is an unbroken curve with no "holes" or "gaps".
Discrete
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 ‘x\begin{align*}x\end{align*}’ may have.
Range
The range of a relation is the set of possible values that ‘y\begin{align*}y\end{align*}’ may have.
Coordinates
The coordinates are the ordered pair (x,y)\begin{align*}(x, y)\end{align*} that represents a point on the Cartesian plane.

### Guided Practice

1. Which relation is discrete? Which relation is 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 (N)\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 ‘x\begin{align*}x\end{align*}’. D={x|3x3,x  R}\begin{align*}D=\{x|-3\le x\le 3, x \ \in \ R\}\end{align*}

The range represents the values of ‘y\begin{align*}y\end{align*}’. R={y|3y3,y  R}\begin{align*}R=\{y|-3 \le y \le 3, y \ \in \ R \}\end{align*}
(ii) D={x|x  R}\begin{align*}D=\{x|x \ \in \ R\}\end{align*}
R={y|4y4,y  R}\begin{align*}R=\{y|-4 \le y \le 4, y \ \in \ 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 discrete. (The salesman can't sell a portion of a computer, so the data points can't be connected.) The number of computers sold and must be whole numbers. The wages must be natural numbers.
A suitable domain is D={x|x0,x  W}\begin{align*}D=\{x|x \ge 0, x \ \in \ W\}\end{align*}
A suitable domain is R={y|y=200+100x,x  N}\begin{align*}R=\{y|y=200+100x, x \ \in \ N\}\end{align*}

### Practice

Use the graph below for #1 and #2.

1. Is the relation discrete, continuous, or neither?
2. Find the domain and range for the relation.

Use the graph below for #3 and #4.

1. Is the relation discrete, continuous, or neither?
2. Find the domain and range for each of the three relations.

Use the graph below for #5 and #6.

1. Is the relation discrete, continuous, or neither?
2. Find the domain and range for the relation.

Use the graph below for #7 and #8.

1. Is the relation discrete, continuous, or neither?
2. Find the domain and range for the relation.

Examine the following pattern.

Number of Cubes (n)\begin{align*}(n)\end{align*} 1 2 3 4 5 ... n\begin{align*}n\end{align*} ... 200
Number of visible faces (f)\begin{align*}(f)\end{align*} 6 10 14
1. Complete the table below the pattern.
2. Is the relation discrete, continuous, or neither?
3. Write a suitable domain and range for the pattern.

Examine the following pattern.

Number of triangles (n)\begin{align*}(n)\end{align*} 1 2 3 4 5 ... n\begin{align*}n\end{align*} ... 100
Number of toothpicks (t)\begin{align*}(t)\end{align*}
1. Complete the table below the pattern.
2. Is the relation discrete, continuous, or neither?
3. Write a suitable domain and range for the pattern.

Examine the following pattern.

Pattern Number (n)\begin{align*}(n)\end{align*} 1 2 3 4 5 ... n\begin{align*}n\end{align*} ... 100
Number of dots (d)\begin{align*}(d)\end{align*}
1. Complete the table below the pattern.
2. Is the relation discrete, continuous, or neither?
3. Write a suitable domain and range for the pattern.

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### Vocabulary Language: English

TermDefinition
Continuous Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.
Coordinates The coordinates of a point represent the point's location on the Cartesian plane. Coordinates are written in ordered pairs: $(x, y)$.
dependent variable The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.
Discrete 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 function is the set of $x$-values for which the function is defined.
Formula A formula is a type of equation that shows the relationship between different variables.
Function A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
independent variable The independent variable is the input variable in an equation or function, commonly represented by $x$.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Range The range of a function is the set of $y$ values for which the function is defined.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

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