# Graphs and Functions

## Relations and Functions


### Find the Domain and Range of a Relation

As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.
When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation.
There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.
A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all x-coordinates is the domain of the relation and the set of all y-coordinates is the range. Generally we write the numbers in ascending order for both the domain and range.

### Determine if a Relation is a Function

A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
The birthday example from  helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24th as their birthday and that June and Liz share August 2nd. Since each person has exactly one birthday, the relation in  is a function.
The relation shown by the graph in  includes the ordered pairs  and  Is that okay in a function? No, as this is like one person having two different birthdays.
In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for y. If each value of x results in only one value of y, then the equation defines a function.

### Find the Value of a Function

It is very convenient to name a function and most often we name it f, g, h, F, G, or H. In any function, for each x-value from the domain we get a corresponding y-value in the range. For the function f, we write this range value y as  This is called function notation and is read f of x or the value of f at x. In this case the parentheses does not indicate multiplication.
We call x the independent variable as it can be any value in the domain. We call y the dependent variable as its value depends on x.
Much as when you first encountered the variable x, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.
Let’s look at the equation  To find the value of y when  we know to substitute  into the equation and then simplify.
The value of the function at  is 3.
We do the same thing using function notation, the equation  can be written as  To find the value when  we write:
The value of the function at  is 3.
This process of finding the value of  for a given value of x is called evaluating the function.
In the last example, we found  for a constant value of x. In the next example, we are asked to find  with values of x that are variables. We still follow the same procedure and substitute the variables in for the x.
Many everyday situations can be modeled using functions.

### Key Concepts

1. Function Notation: For the function 


2. Independent and Dependent Variables: For the function 



### Practice Makes Perfect

Find the Domain and Range of a Relation
In the following exercises, for each relation ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercises, use the mapping of the relation to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function, ⓑ find the domain of the function, and ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function: ⓐ  ⓑ  ⓒ 
In the following exercises, evaluate the function: ⓐ  ⓑ  ⓒ 
In the following exercises, evaluate the function.
In the following exercises, solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?