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Section 3.1
Relations and Functions
Objectives
1. Understanding the Definitions of Relations and Functions
2. Determining if Equations Represent Functions
3. Using Function Notation; Evaluating Functions
4. Using the Vertical Line Test
5. Determining the Domain of a Function Given the Equation
Functions appear all around us and are a fundamental part of mathematics. Functions occur when one quantity
depends on another.
Here are just a few examples:
The revenue generated by selling a certain product depends on the number of items sold.
The volume of a cube depends on the length of an edge of the cube.
For a person who works by the hour, the gross pay depends on the number of hours worked.
If you think about it, you can probably come up with countless situations where one thing, event or occurrence
depends on another. Before we officially define a function, we must first define a relation.
Objective 1: Understanding the Definitions of Relations and Functions
Definition
Relation
A relation is a correspondence between two sets A and B such that each element of set A
corresponds to one or more elements of set B. Set A is called the domain of the relation and set
B is called the range of the relation.
Suppose that Adam, Patrice, and Scott are college roommates and suppose that their weights are 180 lbs, 190
lbs, and 180 lbs respectively. The pairing of the roommates’ names with their weights is a relation. In this
example, we are relating a set of names with a set of weights. The set of names Adam, Patrice, Scott is the
domain of this relation while their weights 180lbs,190lbs is the range. Notice that 180 lbs is listed only once
in the range. There is no need to list an element of either set more than once.
Domain
Range
Adam
180 lbs
Patrice
190 lbs
Relation A
Scott
A relation can be represented as a set of ordered pairs with the elements of the domain listed first. The relation
(A) above can be written as the set of ordered pairs { (Adam, 180 lbs), (Patrice, 190 lbs), (Scott, 180 lbs) }.
Let’s now switch the domain and range of the first relation. The new relation (B) can be written as the set of
ordered pairs {(180 lbs, Adam), (180 lbs, Scott), (190 lbs, Patrice)}.
Domain
Range
180 lbs
Adam
Relation B
190 lbs
Patrice
Scott
Suppose that a delivery man was to deliver a package to the roommate who weighs 180 lbs. Since there are two
people who weigh 180 lbs, there is no way to know for sure who this package belongs to. This Relation B is an
example of a relation that is not well-defined.
A well-defined relation has exactly one output (range) value for any input (domain) value. Well-defined
relations are called functions. Relation A is an example of a well-defined function.
Definition
Function
A function is a relation such that for each element in the domain, there corresponds exactly
one and only one element in the range. In other words, a function is a well-defined relation.
The first relation given is a function because for every name in the domain, there is exactly one corresponding
weight in the range. The second relation is not a function because there is a domain value
(180 lbs) that corresponds to two range values (Adam and Scott).
Note: The elements of the domain and range are typically listed in ascending order when using set
notation.
If the domain and range of a relation are sets of real numbers, then the relation can be represented by plotting
the ordered pairs in the Cartesian plane. The set of all of the x-coordinates is the domain of the relation and the
set of all y-coordinates is the range of the relation.
Objective 2: Determine if Equations Represent Functions
In the two relations described above, we saw that a finite number of ordered pairs in the Cartesian plane can
represent a relation. Relations can also be described by infinitely many ordered pairs in the plane.
To determine if an equation represents a function, we must show that for any value of in the domain, there is
exactly one corresponding value in the range.
Objective 3: Using Function Notation; Evaluating Functions
When an equation is explicitly solved for y, we say that “y is a function of x” or that the variable y depends on
the variable x. Thus, x is the independent variable and y is the dependent variable.
Instead of using the variable y, letters such as f , g or h (and others) are commonly used for functions. For
example, suppose we wanted to name a function f. Then for any x-value in the domain, we will call the y-value
(or function value) f ( x ) . The symbol f ( x ) is read as “the value of the function f at x” or simply “f of x”. For
3
3
example, the function y  x  6 can be written as f ( x)  x  6 . The notation f ( x ) is called function
2
2
notation.
3
x  6 then the expression f ( 4) represents the range value (y-coordinate) when the
2
3
domain value (x-coordinate) is 4 . To find this value, replace x by 4 in the equation f ( x)  x  6 to get
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f (4)  (4)  6  6  6  0 . Thus the ordered pair ( 4, 0) must lie on the graph of f.
2
In addition, when f ( x) 
The symbol f ( x ) does not mean f times x. The notation f ( x ) refers to the value of the function at x.
The expression (1) 2 does not equal 12 .
The expression
f ( x  h)  f ( x )
is called the difference quotient and is very important in calculus.
h
Function notation is not restricted to the use of x for the independent variable or to the use of f for the name of
the function, which is the dependent variable. Other letters commonly used in place of x are h, t, r, a, and b.
Letters such as A for area, V for volume, and d for distance are often used to represent the names of functions.
In fact, any letter can be used to represent values of either the domain or the range.
Objective 4: Using the Vertical Line Test
Although every function has a graph, it is not the case that every graph in the plane represents a function. A
graph does not represent a function if two or more points on the graph with the same first coordinate have
different second coordinates. The vertical line test can be used to quickly determine whether or not a graph
represents a function.
The Vertical Line Test
A graph in the Cartesian plane is the graph of a function if and only if no vertical
line intersects the graph more than once.
Figure 4
Objective 5: Determining the Domain of a Function Given the Equation
The domain of a function y  f ( x) is the set of all values of x for which the function is defined. In other words,
a number x  a is in the domain of a function f if f ( a ) is a real number. For example, the domain of
f ( x)  x 2 is all real numbers since for any real number x  a , the value of f (a )  a 2 is also a real number.
It is very helpful to classify a function to determine its domain. For example, the function f ( x)  x 2 belongs to
a class of functions called polynomial functions. The domain of every polynomial function is all real numbers.
Polynomial functions are discussed in great detail in Chapter 4.
Definition
Polynomial Function
The function f ( x)  an xn  an1xn1  an2 xn2   a1x  a0 is a polynomial function
of degree n where n is a nonnegative integer and a0 , a1 , a2 , , an are real numbers.
The domain of every polynomial function is  ,   .
Many functions can have restricted domains. For example, the quotient of two polynomial functions is called a
x5
rational function. The rational function f ( x) 
is defined everywhere except when x  4 because the
x4
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value f (4)  is undefined. Therefore, the domain of a rational function consists of all real numbers for
0
which the denominator does not equal zero. We will investigate rational functions further in Chapter 4.
Definition
Rational Function
A rational function is a function of the form f ( x) 
g ( x)
where g and h are polynomial
h( x )
functions such that h( x)  0.
The domain of a rational function is the set of all real numbers such that h( x)  0 .
Root functions can also have restricted domains. Consider the root function f ( x)  x  1 . The number
x  3 is in the domain because f (3)  3  1  4  2 is a real number. However, 5 is not in the domain
because f (5)  5  1  4  2i is not a real number. Therefore, the domain of f ( x)  x  1 consists of
all values of x for which the radicand is greater than or equal to zero. The domain of g is the solution to the
inequality x  1  0 .
x 1  0
x  1 Subtract 1 from both sides.
Therefore, the domain of f is  1,   . Root functions with roots that are odd numbers such as 3 or 5 can have
negative radicands. Therefore, the domain of a root function of the form f ( x)  n g ( x) where n is an odd
positive integer consists of all real numbers for which g ( x ) is defined.
Definition
Root Function
The function f ( x)  n g ( x) is a root function where n is a positive integer.
1.
2.
If n is even, the domain is the solution to the inequality g ( x)  0 .
If n is odd, the domain is the set of all real numbers for which g ( x ) is defined.
Below is a quick guide for finding the domain of 3 specific types of functions:
Class of function
Form
Domain
Polynomial Functions
f ( x)  an xn  an1 xn1 
Rational Functions
f ( x) 
Root Functions
f ( x)  n g ( x) , where g ( x) is a function and n is a
 a1 x  a0
g ( x)
where g and h  0 are polynomial functions
h( x )
positive integer
Domain is  ,  
Domain is all real numbers such that
h( x)  0
1. If n is even, the domain is the solution
to the inequality g ( x)  0 .
2. If n is odd, the domain is the set of all
real numbers for which g is defined.
We will study other classes of functions later in this text. For now, we limit our discussion to polynomial
functions, rational functions and root function.