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?

3. What is the range of the function?  

Concept Summary  

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.