Lesson 2.1: Domains, Ranges, and Compositions of Functions
A relation is a relationship between two variables that results in a set of ordered pairs. A
is a special kind of relation: it is a relationship between two variables such that each value of the first
variable is paired with
value of the second variable.
The
of a function is the set of all possible values of the first variable. The
of a function is the set of all possible values of the second variable.
You can also use the
to determine if a graph represents a function: if
every vertical line intersects a given graph at no more than one point, then the graph represents a
function.
e.g. Which of the following relations are functions? Explain.
a)
 2,3 ,  4,1 ,  0,1 ,  1, 2
b)
c)
5, 2 , 1,3 , 0, 4 , 5,1
d)
Relations and functions can also be described using mapping diagrams that illustrate the pairing
between each domain element and its corresponding range element(s). For example, the relation
5, 2 , 1,3 , 0, 4 , 5,1 could be described as:
5
-2
3
1
0
-4
1
If there is a correspondence between values of the domain, x , and values of the range, y , that is a


function, then y  f  x  , and  x, y  can be written as x, f  x  . The notations f  x  and f : x are
“. The number represented by f  x  is the value of the function f at x . For
read as “
example, a function f that is mapping any domain element x to a range element that is equal to
2x  1 can be notated as either f  x   2 x  1 or f : x  2 x  1 . The variable x is known as the
variable, while the variable y , or f  x  , is known as the
variable.
e.g. Given the function g : x  x 2  3x , evaluate:
a) g  1
c) g  2a  1
b) g : 2
e.g. Given the function h  x   1  4 x , find x when:
b) h  x   3a  7
a) h : x  9
e.g. Which of the following relations are functions? Explain.
a)
b)
5
-2
5
3
1
0
c) y   x  2
1
-4
1
0
d) y 
1 3
x  2x  3
2
-2
3
Lesson 2.1 (continued)
Many relations have restricted domains and/or ranges that can be expressed in set or interval notation.
A collection of individual points is known as a
relation, while a line, ray, segment, or
smooth curve is known as a
relation.
e.g. State the domain and range for each of the following relations:
a)
b)
5
-2
3
1
-4
1
0
c)
 0,3 ,  2, 4 ,  2,1 , 3,3
d) f  x   
1
x3
2
When you apply a function rule onto the result of another function rule, you are performing a
of functions. The composition of f with g , denoted f



g or  f g  x  , is

defined by f g  x  . The domain of y  f g  x  is the set of domain values of g whose range
values are in the domain of f . The function f

g is called the composite function of f with g . To

evaluate a composition of functions such as f g 1 , you have a choice of two methods:
1. you could start by evaluating g 1 , and then substitute the result into the function f , OR


2. you could start by finding an expression for f g  x  , and then substitute x  1 into this
composite function
e.g. Given f  x   4 x  3 and g  x   x 2  1 , find:

a) f g  3
c) f
f
e) f
g

f
Note from the last two examples above that f
compositions of functions are not usually
b)
g
f  2


d) g f  a  1
f) g
f
g  g f . In fact, this is usually the case meaning that
. However, there are some exceptions.
e.g. Find two functions f and g whose composite functions f
g and g f are equivalent.