Math 106
Section 2.4 Functions, Function Notation, and the Graph of a Function
Defn: A function is a relation that pairs each element from the domain with exactly one element from the
range. [every element in the domain must be used exactly once]
Domain
x-values
input
independent variable
Range
y-values
output
dependent variable
EX: Determine whether each of the following relations is a function. If it is not a relation, state why.
a.
b.
Joe
Joe
Job 1
Job 1
Bob
Bob
Job 2
Job 2
Larry
Larry
c.
d.
Job 1
Joe
Job 1
Joe
Job 2
Job 2
Bob
Bob
Job 3
Larry
Job 3
Larry
Job 4
e.
Job 4
f.
Input
Output
0
3
g. x 2  2 y  4
3
3
6
3
Input
Output
9
3
h.
y  x 3
0
4
1
5
1
6
2
7
Vertical Line Test: A given graph is the graph of a function, if and only if every vertical line intersects the
graph in at most one point.
Implied domain: all x-values for which the function represents a real number
EX: Use the VLT to determine whether each graph is the graph of a function. State the domain and range of
each relation.
6
6
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
-6
-6
Function? _______________
Domain _________________
Range___________________
Function? _______________
Domain _________________
Range___________________
6
6
4
4
2
2
-5
5
-2
-4
-6
Function? _______________
Domain _________________
Range___________________
-5
5
-2
-4
-6
Function? _______________
Domain _________________
Range___________________
EX: Use the function f x   3x  2 to determine the following values.
a.
f 0 
b.
f 6 
c.
f  2 
d.
f t  
e.
f  
f.
f a  5 
g. g.
f x  h  
h. h.
f x  h   f x 

h
EX: Use the function g x    x 2  6 x  5 to determine the following values.
a.
g. g x  h 
g 3 
b. g 1 
c. 2 g x  
h.
d. g 2 x  
e.
g x   g h 
f.
g x   h 
g x  h   g x 

h
EX: Given the graph of f x  , determine the values below.
(-4, 4)
f 0 
6
f(x):
(-2, 4)
f  4 
4
(2, 2)
2
-5
(0, 0)
f 2 
5
f  3 
-2
-4
-6
f 0  f  2

02
EX: Determine the (implied) domain of each function.
a.
f x  
x5
x 3
b. g x   2 x  1
c. hx  
d.
j x  
x2  3
4
2 x
x2  9
e. k x   16  x 2
f.
l x  
x
x5