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# Relations and Functions

## Key Concepts

• Relation and function.
• Domain and range definition.
• Continuous and discrete.
• A function as one-to-one or not one-to-one.
• Constraints on the domain.

### Recognize domain and range

#### Relation

A relation is any set of ordered pairs.

#### Function

A function is a relation in which each input is assigned to exactly one output.

#### Domain

The domain of the function is the set of inputs.

#### Range

The range of the function is the set of outputs.

So, by convention, inputs are x−values

and outputs are y−values

Consider a tabular column:

Here the domain of this function is the set of x−values , {1, 2, 3, 4, 5}

The range of this function is the set of y−values, {2, 4, 6, 8}

Example 1:

What are the domain and the range of the function?

Solution 1:

The domain of a function is the set of inputs.

The range of a function is the set of outputs.

The domain of this function is the set of x−values, {−3, −1, 1, 2, 3}.

The range of this function is the set of y−values, {−3, −2, 0, 1}.

Example 2:

What are the domain and the range of the function?

Solution 2:

The domain of a function is the set of inputs.

The range of a function is the set of outputs.

The domain of this function is the set of x−values: {−3, −1, 0, 1}.

The range of this function is the set of y−values: {−2, −1, 3}.

### Analyze reasonable domains and ranges

Q: What is a reasonable domain and range of function as mentioned in the situation given below?

Situation A:

A hose fills a 10,000-gallons swimming pool at a rate of 10 gallons per minute.

Solution:

A reasonable domain is from 0 minutes to the time it takes to fill the pool.

A reasonable range is from 0 to 10,000 gallons, the capacity of the pool.

Situation B:

A school needs to order chairs for its tables. One table can accommodate two chairs.

Solution:

A reasonable domain is from 0 tables to the number of tables needed.

A reasonable range is multiples of 2 from 0 to 2 times the number of tables needed.

Analyze the function of situation A with the graph.

The volume of water in the pool can be determined at any point in time, for any value of x.

Here, the domain of a function is continuous.

Analyze the function of situation B with the graph.

The number of chairs and tables must be a whole number. There cannot be parts of tables or chairs. Here, the domain of a function is discrete.

### Continuous function

The domain of a function is continuous when it includes all real numbers.

The graph of the function is a line or curve

### Discrete function

The domain of a function is discrete when it consists of just whole numbers or integers.

The graph of the function is a series of data points.

### Classify relations and functions

#### One-to-one

A function is one-to-one if no two elements of the domain map to the same element in the range.

#### Not one-to-one

When two or more elements of the domain map to the same element of the range, the function is not one-to-one.

Example 1:

{(1,2), (3, 4), (5, 6), (7,8)}

Solution:

The relation is a function. Every element of the domain {1, 3,5,7} maps to exactly one element of the range {2, 4, 6, 8}.

Since none of the range values are shared, the function is one-to-one.

Example 2:

Solution:

The relation is a function.

Every element of the domain maps to exactly one element of the range.

Since more than one element in the domain maps to a single element in the range.

So, the function is not one-to-one.

Example 3: Is each relation a function? If so, is it one-to-one or not one-to-one?

Solution:

The relation is not a function.

The elements of the domain maps to one or more elements of the range.

Example 4: Is each relation a function? If so, is it one-to-one or not one-to-one?

Solution:

A function is one-to-one if no two elements of the domain map to

the same element in the range.

{(1, 3), (2, 2), (3, 1), (4, 0)}

The relation is a function.

Every element of the domain {1, 2,3,4} maps to exactly one element of the range {3, 2, 1, 0}.

Since none of the range values are shared, the function is one-to-one.

### Identify constraints on the domain:

The diagram shows shipping charges as a function of the weight of several online orders. Based on the situation, what constraints, if any, are on the domain of the function?

An order must have a weight greater than zero, so the domain of the function is confined to values greater than 0.

## Exercise

1. Fill in the blanks:
1. A __________ is any set of ordered pairs.
2. The __________ of a function is the set of inputs.
3. Range is the set of ___-values.
4. A function is ________if no two elements of the domain map to the same element in the range.
5. The domain of a function is __________when it includes consists of just whole numbers or integers.
2. What is the domain of the function?
1. What is the range of the function?

### What we have learned

• Determine whether a relation is a function.
• Recognize domain and range.
• Analyze reasonable domains and ranges
• Classify relations and functions.
• Identify constraints on the domain.

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