Relations, Functions, and Function Notation

Definition of a Relation, Domain, and Range

Examples

1. Consider the relation that sends a student to that student's age.

2. Consider the relation that sends a student to the courses that student is taking.

3. Consider the relation that sends a parent to the parent's child.

4. Consider the relation that sends a key word either to its matches from the Yahoo search engine or to the statement "No matches found."

Each of these are examples of relations.  The definition below formalizes this idea.

 Definition: A relation is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned one or more elements of the range.

Remark:  To define a relation three things must be designated:  the range set, the domain set and the rule of assignment.

Non-Example

Let the domain be the set of all LTCC students and the range be the set of all math course offerings at LTCC.  Then the map that takes a student and sends the student to the math course he or she is taking is not a relation since there are students who are not taking math courses.

Example

(2,3), (2,4), (3,7), and (5,2)

defines a relation with

Domain:  {2,3,5}

Range:  {2,3,4,7}

Example

A circle represents the graph of a relation with domain consisting of x-values from the left side of the circle to the right side.  The range consists of y-values from the bottom to the top.

Functions

A function is a special type of relation where every input has a unique output.

 Definition:   A function is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned exactly one element of the range.

Example

(3,3), (4,3), (2,1), (6,5)

is a function with

Domain: {2,3,4,6}

Range: {1,3,5}

Non-Example:

(2,1), (5,6), (2,3), (6,7)

is not a function since 2 gets sent to more than one value.

The Vertical Line Test

To determine whether a graph is the graph of a function, we use the following criterion.

 If any vertical line passes through a graph at more than one point, then the graph is not the graph of a function.  Otherwise it is the graph of a function.*

* For a more precise definition click here

Example:

A circle is not the graph of a function as shown below. Example:

A (non-vertical) line is the graph of a function. Function Notation

 Definition We write f (x) to mean the function whose input is x.

Example:

If

f(x)  =  2x - 3

then

f(4)  =  2(4) - 3  =  5

We can think of f as the function that takes the input multiplies it by 2 and subtracts 3.  Sometimes it is convenient to write f(x) without the x. Thus:

f( ) = 2( ) - 3

whatever is in the parentheses, we put inside.  For example:

f(x - 1)  =  2(x - 1) - 3

and

f(x + 4) - f(x)               [2(x + 4) - 3] - [2(x) - 3]
=
4                                    4

2x + 8 - 3 - 2x + 3
=                                            Distributing the 2 and the -

4

8
=             = 2                                Combining Like Terms

4

Example

If

f(x)  =  7

then

f(12)  =  7

Here f is called the constant function.  Whatever comes in to f, the number 7 comes out.

Function Arithmetic

We define the sum, difference, product and quotient of functions in the obvious way.

Example

If

x + 1
f(x)  =
x - 1

and

g(x) = x2 + 4

then

x + 1
(f + g)(x) =                   + (x2 + 4)
x - 1

x + 1
(f - g)(x) =                   - (x2 + 4)
x - 1

x + 1
(f g)(x) =                 (x2 + 4)
x - 1

and 