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# Domain and Range of a Function

## Discrete and continuous functions and dependent and independent values

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Recognizing Functions

Tammy has opened an ice cream stand to earn money for her college tuition. She records the number of cones she sells each day in a table like the one shown below.

 Days in operation Number of cones sold 1 48 2 56 3 77 4 79 5 82 6 98 7 105

Tammy is very pleased with the numbers of cones sold for her first week in operation. Now she wants to analyze the results shown in the table. What information can Tammy record from her table?

In this concept, you will learn to recognize functions.

### Functions

Many things in the real world consist of a relationship between two things. The number of hours a person works and the amount of money earned, the number of gallons of gas bought and the distance a person can drive and the names of the students in a class and their heights are just a few examples of relationships between two things.

The names of the students in a class paired with their heights is an example of a relation. These two things can be paired as (name, height) or as (height, name). The name and the height are written as an ordered pair which simply means that one comes first and the other comes second. When a relationship is expressed as an ordered pair it is called a relation. A set of ordered pairs is a relation.

If the relationship between the names of the students and their heights is written as (name, height) with the name as the first thing in the ordered pair, then the names are the set of all staring points. This set of starting points is called the domain of the relation. The heights are the second thing in the ordered pair and are the set of all ending points. This set of ending points is called the range of the relation.

If the variable ‘x\begin{align*}x\end{align*}’ represents the names of the students and the variable ‘y\begin{align*}y\end{align*}’ represents the heights of the students, then the ordered pair (name, height) can be written as (x,y)\begin{align*}(x,y)\end{align*}. The set of values for ‘x\begin{align*}x\end{align*}’ will be the domain and the set of values for ‘y\begin{align*}y\end{align*}’ will be the range.

Look at the following set of ordered pairs:

(1,4),(2,6),(2,8),(3,10),(4,12)\begin{align*}(1,4),(2,6),(2,8),(3,10),(4,12)\end{align*}

The values are written as ordered pairs (x,y)\begin{align*}(x,y)\end{align*} such that the domain includes the values (1,2,3,4)\begin{align*}(1,2,3,4)\end{align*} and the range includes the values (4,6,8,10,12)\begin{align*}(4,6,8,10,12)\end{align*}. Notice that the x\begin{align*}x\end{align*}-value of 2 is paired with two different y\begin{align*}y\end{align*}-values. A relation can have a member of the domain paired with more than one member from the range.

Look at this set of ordered pairs:

(1,3),(2,5),(3,7),(4,9),(5,11)\begin{align*}(1,3),(2,5),(3,7),(4,9),(5,11)\end{align*}

The values are written as ordered pairs (x,y)\begin{align*}(x,y)\end{align*} such that the domain includes the values (1,2,3,4,5)\begin{align*}(1,2,3,4,5)\end{align*} and the range includes the values (3,5,7,9,11)\begin{align*}(3,5,7,9,11)\end{align*}. Notice that each x\begin{align*}x\end{align*}-value is paired with only one y\begin{align*}y\end{align*}-value. A relation such that each member of the domain is paired with one and only one member of the range is called a function.

### Examples

#### Example 1

Earlier, you were given a problem about Tammy and her ice cream cones. She wants to record information from the table.

Tammy can record the domain and range from the table and determine if the information represents a relation or a function.

First, make a list of ordered pairs to represent the information in the table.

{(1,48),(2,56),(3,77),(4,79),(5,82),(6,98),(7,105)}\begin{align*}\{(1,48),(2,56),(3,77),(4,79),(5,82),(6,98),(7,105) \}\end{align*}

Next, list the members of the domain. Remember to list each x\begin{align*}x\end{align*}-value only once.

(1,2,3,4,5,6,7)\begin{align*}(1,2,3,4,5,6,7)\end{align*}

Next, list the members of the domain. Remember to list each y\begin{align*}y\end{align*}-value only once.

(48,56,77,79,82,98,105)\begin{align*}(48,56,77,79,82,98,105)\end{align*}

Next, look at the ordered pairs to see if any x\begin{align*}x\end{align*}-value is paired with more than one y\begin{align*}y\end{align*}-value.

Every x\begin{align*}x\end{align*}-value from the domain is paired with only one y\begin{align*}y\end{align*}-values from the range.

Then, state if the ordered pairs represent a relation or a function.

A function.

#### Example 2

Determine if the following represents a function or a relation.

First, follow each arrow from its value in the domain to its value in the range.

Next, write a set of ordered pairs to represent the arrows. Remember the domain represents the x\begin{align*}x\end{align*}-values and the range represents the y\begin{align*}y\end{align*}-values.

(3,4),(2,8),(1,0),(0,8),(1,4)\begin{align*}(-3, 4),(-2, 8),(-1, 0),(0, 8),(1, -4)\end{align*}

Next, check to see if each x\begin{align*}x\end{align*}-value is paired with one y\begin{align*}y\end{align*}-value.

Then, state if the ordered pairs represent a relation or a function.

A function.

#### Example 3

Do the following ordered pairs represent a relation or a function?

{(0,5),(1,6),(2,7),(1,8),(3,9),(2,10)}\begin{align*}\{ (0,5),(1,6),(2,7),(1,8),(3,9),(2,10) \}\end{align*}

First, list the members of the domain. Remember to list each x\begin{align*}x\end{align*}-value only once.

(0,1,2,3)\begin{align*}(0,1,2,3)\end{align*}

Next, list the members of the domain. Remember to list each y\begin{align*}y\end{align*}-value only once.

(5,6,7,8,9,10)\begin{align*}(5,6,7,8,9,10)\end{align*}

Next, look at the ordered pairs to see if any x\begin{align*}x\end{align*}-value is paired with more than one y\begin{align*}y\end{align*}-value.

(1,6)\begin{align*}(1,6)\end{align*} and (1,8)\begin{align*}(1,8)\end{align*}. The x\begin{align*}x\end{align*}-value of 1 is paired with two different y\begin{align*}y\end{align*}-values 6 and 8.

(2,7)\begin{align*}(2,7)\end{align*} and (2,10)\begin{align*}(2,10)\end{align*}. The x\begin{align*}x\end{align*}-value of 2 is paired with two different y\begin{align*}y\end{align*}-values 7 and 10.

Then, state if the ordered pairs represent a relation or a function.

A relation.

#### Example 4

Do the following ordered pairs represent a relation or a function?

{(0,5),(1,5),(2,5),(3,5)}\begin{align*}\{ (0,5),(1,5),(2,5),(3,5) \}\end{align*}

First, list the members of the domain. Remember to list each x\begin{align*}x\end{align*}-value only once.

(0,1,2,3)\begin{align*}(0,1,2,3)\end{align*}

Next, list the members of the domain. Remember to list each y\begin{align*}y\end{align*}-value only once.

\begin{align*}(5)\end{align*}

Next, look at the ordered pairs to see if any \begin{align*}x\end{align*}-value is paired with more than one \begin{align*}y\end{align*}-value.

Each \begin{align*}x\end{align*}-value is paired with only one \begin{align*}y\end{align*}-value.

Then, state if the ordered pairs represent a relation or a function.

A function.

#### Example 5

Do the following ordered pairs represent a relation or a function?

\begin{align*}\{(8, -2), (5, -3), (0, -9), (8, -4)\}\end{align*}

First, list the members of the domain. Remember to list each \begin{align*}x\end{align*}-value only once.

\begin{align*}(0,5,8)\end{align*}

Next, list the members of the domain. Remember to list each \begin{align*}y\end{align*}-value only once.

\begin{align*}(-9,-4,-3,-2)\end{align*}

Next, look at the ordered pairs to see if any \begin{align*}x\end{align*}-value is paired with more than one \begin{align*}y\end{align*}-value.

The \begin{align*}x\end{align*}-value of 8 is paired with two \begin{align*}y\end{align*}-values -2 and -4.

Then, state if the ordered pairs represent a relation or a function.

A relation.

### Review

Are the relations functions? Write function if it is a function and not a function if it is not a function.

1. \begin{align*}\{(4, 7), (8, 11), (4, 9), (8, 13) \}\end{align*}
2. \begin{align*}\{(2, 7), (2, 11), (4, 12), (8, 13) \}\end{align*}
3. \begin{align*}\{(3, 4), (5, 6), (7, 8), (8, 10) \}\end{align*}
4. \begin{align*}\{(12, 7), (11, 11), (14, 9), (18, 13) \}\end{align*}
5. \begin{align*}\{(3, 7), (4, 11), (3, 9), (12, 13) \}\end{align*}
6. \begin{align*}\{(8, 7), (9, 6), (10, 5), (11, 4) \}\end{align*}
7. \begin{align*}\{(4, 2), (8, 1), (3, 9), (8, 7) \}\end{align*}
8. \begin{align*}\{(11, 17), (18, 21), (14, 19), (18, 13) \}\end{align*}
9. \begin{align*}\{(4, 7), (8, 11), (4, 9), (8, 13) \}\end{align*}
10. \begin{align*}\{(-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 1) \}\end{align*}
11. \begin{align*}\{(6, 25), (12, 35), (18, 45), (24, 55) \}\end{align*}
12. \begin{align*}\{(2, 4), (3, 5) (2, 6), (7, 9) \}\end{align*}
13. The amount of bananas you buy at a store for $.85 per pound. 14. The amount of carrots that you buy at a store for$.29 per pound.
15. The steady price increase of a bus ticket over time.

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Color Highlighted Text Notes

### Vocabulary Language: English

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.

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.

Relation

A relation is any set of ordered pairs $(x, y)$. A relation can have more than one output for a given input.