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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

Do you know how to identify a function? Have you ever volunteered at a soup kitchen? Take a look at this dilemma.

A soup kitchen prepares food for people every day of the month. The supervisor keeps count of the number of people who eat every day. Her data table for the first few days is below.

Day of Month # of Visitors
1 82
2 84
3 87
4 80
5 91
6 93
7 104
8 84
9 88

Can you identify the range of this data? The domain of the data? Is this set of data a function?

This Concept will teach you about relations, ranges, domains and functions. You will know how to answer these questions by the end of the Concept.

### Guidance

Many numbers have precise and predictable relationships—the number of motorcycles and the number of tires, the numbers of hours you work and the money you get paid, the number of years you go to college and your lifetime earnings.

Here, you will learn to recognize a relation as a set of ordered pairs that relates an input to an output, and a function as relation for which there is exactly one output for each input, using tables.

When we work with relations and functions, we work with the world of relationships. We look at how one factor impacts or effects another factor.

What is a relation?

A relation is written as a set of ordered pairs where one value is equal to \begin{align*}x\end{align*} and one value is equal to \begin{align*}y\end{align*}.

With a relation, we are looking at the relationship between one factor and another.

Take a look.

A motorcycle has an ordered pair of bikes to tires as (1, 2).

This means that for every one motorcycle there are two tires. This is a relation.

A relation is a set of ordered pairs.

The first coordinate would be the number of motorcyles and the second coordinate would be the number of tires.

There are parts of a relation too. We can have a domain and a range for every relation. The values in the domain and range help us to understand the relation.

The domain is composed of the values in first column or the \begin{align*}x\end{align*} coordinate in the relation.

The range is composed of the second column or the \begin{align*}y\end{align*} value of the relation.

There are different types of relations too. A relation can be a function or not a function.

A function is a relation in which each member of the domain is paired with exactly one member of the range.

In other words, a number in the domain cannot have two values for the range. When we look at the values in the domain and the range, we can figure out if the relation is a function or not.

Write the definitions for relation, domain, range and function in your notebook.

Now let's apply this information.

Is this relation a function?

To figure this out, we have to compare the values from the first column with the values in the second column. For each value of the domain, there is exactly one value in the range. In other words, there aren’t any values that repeat.

Therefore, this relation is a function.

Here is another one.

Is this relation a function?

This relation is not a function because 12 in the domain is paired with two values in the range. Notice that we could write these values in ordered pairs as well. Looking at a set of ordered pairs can also help us determine whether or not a relation is a function.

Identify whether or not each relation is a function.

#### Example A

\begin{align*}(1,2)(2,6)(7,9)(8,4)\end{align*}

Solution: Function

#### Example B

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

Solution: Not a function

#### Example C

\begin{align*}(1,11)(2,12)(3,16)(1,14)\end{align*}

Solution: Not a function

Now let's go back to the dilemma at the beginning of the Concept.

We can rewrite this data as a relation, a set of ordered pairs. The first coordinate would be the day of the month and the second coordinate would be the number of visitors.

The relation would look like this {(1, 82), (2, 84), (3, 87), (4, 80), (5, 91), (6, 93), (7, 104), (8, 84), (9, 88)}. Notice that the days of the week form the \begin{align*}x\end{align*} value and the number of visitors forms the \begin{align*}y\end{align*} value.

The braces, {}, indicate that these are all the ordered pairs in the set.

We can have a domain and a range for every relation.The values in the domain and range help us to understand the relation.

The domain is made up of the values in first column or the \begin{align*}x\end{align*} coordinate in the relation.

The range is made up of the second column or the \begin{align*}y\end{align*} value of the relation.

A function is a relation in which each member of the domain is paired with exactly one member of the range.

In other words, a number in the domain cannot have two values for the range.

In this chart, every day of the month has only one number of visitors. Therefore, this relation is a function.

### Vocabulary

Relation
a set of ordered pairs.
Domain
the \begin{align*}x\end{align*} value in a table or function.
Range
the \begin{align*}y\end{align*} value in a table or function.
Function
Each value in the domain is connected to only one value in the range.

### Guided Practice

Here is one for you to try on your own.

Is this relation a function? {(8, -2), (5, -3), (0, -9), (8, -4)}

Solution

To figure this out, we look at the values in the domain. The value 8 has two values in the range that are matched with it, so this relations is not a function.

### Practice

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

1. {(4, 7), (8, 11), (4, 9), (8, 13)}
2. {(2, 7), (2, 11), (4, 12), (8, 13)}
3. {(3, 4), (5, 6), (7, 8), (8, 10)}
4. {(12, 7), (11, 11), (14, 9), (18, 13)}
5. {(3, 7), (4, 11), (3, 9), (12, 13)}
6. {(8, 7), (9, 6), (10, 5), (11, 4)}
7. {(4, 2), (8, 1), (3, 9), (8, 7)}
8. {(11, 17), (18, 21), (14, 19), (18, 13)}
9. {(4, 7), (8, 11), (4, 9), (8, 13)}
10. {(-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 1)}
11. {(6, 25), (12, 35), (18, 45), (24, 55)}
12. {(2, 4), (3, 5) (2, 6), (7, 9)}
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.

### Vocabulary Language: English

Continuous

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

dependent variable

The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.
Discrete

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

domain

The domain of a function is the set of $x$-values for which the function is defined.
Formula

Formula

A formula is a type of equation that shows the relationship between different variables.
Function

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

independent variable

The independent variable is the input variable in an equation or function, commonly represented by $x$.
Integer

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

Range

The range of a function is the set of $y$ values for which the function is defined.
Real Number

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

Relation

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

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