* Determine whether a relation represents a function.
* Find the value of a function.
* Determine whether a function is one-to-one.
* Use the vertical line test to identify functions.
* Graph the functions listed in the library of functions.
2. Concepts and Objectives
β« Objectives for this section are:
β« Determine whether a relation represents a function.
β« Find the value of a function.
β« Determine whether a function is one-to-one.
β« Use the vertical line test to identify functions.
β« Graph the functions listed in the library of functions.
3. Functions
β« A relation is a set of ordered pairs.
β« A function is a relation in which, for each distinct value of
the first component of the ordered pairs, there is exactly
one value of the second component.
β« More formally:
If A and B are sets, then a function f from A to B
(written f: A β B)
is a rule that assigns to each element of A
a unique element of set B.
4. Functions (cont.)
β« The set A is called the domain of the function f
β« Every element of A must be included in the function.
β« The set B is called the codomain of f
β« The subset of B consisting of those elements that are
images under the function f is called the range.
β« The range and the codomain may or may not be the
same.
5. Functions (cont.)
β« In terms of ordered pairs, a function is the set of ordered
pairs (A, f (A)).
β« Historical note: The notation f (x) for a function of a
variable quantity x was introduced in 1748 by Leonhard
Euler in his text Algebra, which was the forerunner of
todayβs algebra texts. Many other mathematical symbols
in use today (such as e and ο°) were introduced by Euler
in his writings.
7. Functions (cont.)
β« Example: Decide whether each relation defines a
function.
β« F and H are functions, because for each different
x-value, there is exactly one y-value.
β« G is not a function, because one x-value corresponds
to more than one y-value.
( ) ( ) ( )
ο» ο½
( ) ( ) ( ) ( )
ο» ο½
( ) ( ) ( )
ο» ο½
1,2 , 2,4 , 3, 1
1,1 , 1,2 , 1,3 , 2,3
4,1 , 3,1 , 2,0
F
G
H
= β β
=
= β β β
8. Functions (cont.)
β« Relations and functions can also be expressed as a
correspondence or mapping from one set to another.
β« Note that H is a function, since the x-values donβt repeat,
even if the y-values do.
β2
1
3
β1
2
4
F
x-values y-values
F is a function.
1
2
1
2
3
G
x-values y-values
G is not a function.
9. Domain and Range
Example: Give the domain and range of each relation.
Determine whether the relation defines a function.
a)
b)
( ) ( ) ( ) ( )
ο» ο½
3, 1 , 4,2 , 4,5 , 6,8
β
β3
4
6
7
100
200
300
10. Domain and Range
Example: Give the domain and range of each relation.
Determine whether the relation defines a function.
a)
b)
( ) ( ) ( ) ( )
ο» ο½
3, 1 , 4,2 , 4,5 , 6,8
β
β3
4
6
7
100
200
300
D: {3, 4, 6}; R: {β1, 2, 5, 8}
not a function (the 4s
repeat)
D: {β3, 4, 6, 7};
R: {100, 200, 300}
function
11. Domain and Range From Graphs
Example: Give the domain and range of each relation.
a)
12. Domain and Range From Graphs
Example: Give the domain and range of each relation.
a)
Domain: (βο₯, ο₯)
Range: (βο₯, ο₯)
13. Domain and Range From Graphs
Example: Give the domain and range of each relation.
b)
14. Domain and Range From Graphs
Example: Give the domain and range of each relation.
b)
Domain: [β4, 4]
Domain
15. Domain and Range From Graphs
Example: Give the domain and range of each relation.
b)
Domain: [β4, 4]
Range: [β6, 6]
Domain
Range
16. Domain and Range From Graphs
Example: Give the domain and range of each relation.
c)
17. Domain and Range From Graphs
Example: Give the domain and range of each relation.
c)
Domain: (βο₯, ο₯)
18. Domain and Range From Graphs
Example: Give the domain and range of each relation.
c)
Domain: (βο₯, ο₯)
Range: [β3, ο₯)
19. The Vertical Line Test
β« Graphs (a) and (c) are relations that are functions β that
is, each x-value corresponds to exactly one y-value.
Since each value of x leads to only one value of y in a
function, any vertical line drawn through the graph of a
function must intersect the graph in at most one point.
This is the vertical line test for a function.
β« Graph (b) is not a function because a vertical line
intersects the graph at more than one point.
β« A practical way to test this is to move your pencil or pen
across the graph. If it touches the graph in more than
one place, itβs not a function.
21. Identifying Functions
Example: Decide whether each relation defines a function
and give the domain and range.
a) y = x + 4
Since each value of x corresponds to one value of y, this
is a function. There are no restrictions on x, so the
domain is (βο₯, ο₯). The value of y is always 4 greater
than x, so the range is also (βο₯, ο₯).
25. Identifying Functions
Example: Decide whether each relation defines a function
and give the domain and range.
c)
If we look at the graph of this relation, we can see that it
fails the vertical line test, so it is not a function.
2
y x
=
26. Identifying Functions
Example: Decide whether each relation defines a function
and give the domain and range.
c)
If we look at the graph of this relation, we can see that it
fails the vertical line test, so it is not a function.
Even without the graph, we can see
that the ordered pairs (4, 2) and
(4, β2) both satisfy the equation.
Domain: [0, ο₯)
Range: (βο₯, ο₯)
2
y x
=
27. Function Notation
β« When a function f is defined with a rule or an equation
using x and y for the independent and dependent
variables, we say βy is a function of xβ to emphasize that
y depends on x. We use the notation
called function notation, to express this.
β« We usually read this as βy = f of xβ.
( ),
y f x
=
28. Function Notation (cont.)
β« For the most part, we use f (x) and y interchangeably to
denote a function of x, but there are some subtle
differences.
β« y is the output variable, while f (x) is the rule that
produces the output variable.
β« An equation with two variables, x and y, may not be a
function at all.
Example: is a circle, but not a function
+ =
2 2
4
x y
29. One-to-One Functions
β« In a one-to-one function, each x-value corresponds to
only one y-value, and each y-value corresponds to only
one x-value. In a 1-1 function, neither the x nor the y can
repeat.
β« We can also say that f (a) = f (b) implies a = b.
A function is a one-to-one function if, for
elements a and b in the domain of f,
a β b implies f (a) β f (b).
31. One-to-One Functions (cont.)
β« Example: Decide whether is one-to-one.
We want to show that f (a) = f (b) implies that a = b:
Therefore, f is a one-to-one function.
( )= β +
3 7
f x x
( ) ( )
f a f b
=
3 7 3 7
a b
β + = β +
3 3
a b
β = β
a b
=
33. One-to-One Functions (cont.)
β« Example: Decide whether is one-to-one.
This time, we will try plugging in different values:
Although 3 β β3, f (3) does equal f (β3). This means that
the function is not one-to-one by the definition.
( ) 2
2
f x x
= +
( ) 2
3 3 2 11
f = + =
( ) ( )
2
3 3 2 11
f β = β + =
34. One-to-One Functions (cont.)
β« Another way to identify whether a function is one-to-one
is to use the horizontal line test, which says that if any
horizontal line intersects the graph of a function in more
than one point, then the function is not one-to-one.
β’
β’
one-to-one not one-to-one
35. βToolkit Functionsβ
β« When working with functions, it is helpful to have a base
set of building-block elements.
β« We call these our βtoolkit functions,β which form a set of
basic named functions for which we know the graph,
formula, and special properties.
β« For these definitions we will use x as the input variable
and y = f(x) as the output variable.
β« We will see these functions and their combinations and
transformations throughout this course, so it will be
very helpful if you can recognize them quickly.
36. Constant Function f(x) = c
β« f(x) = c is constant on its entire domain, [c, c].
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: [c, c]
x y
β2 c
β1 c
0 c
1 c
2 c
c
37. Identity Function f(x) = x
β« f(x) = x is increasing on its entire domain, (ββο¬ β).
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: (ββο¬ β)
x y
β2 β2
β1 β1
0 0
1 1
2 2
38. Absolute Value Function
β« decreases on the interval (ββο¬ 0] and
increases on the interval [0, β).
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: (ββο¬ β)
x y
-2 2
-1 1
0 0
1 1
2 2
( )
f x x
=
( )
f x x
=
39. Quadratic Function f(x) = x2
β« f(x) = x2 decreases on the interval (ββο¬ 0] and increases
on the interval [0, β).
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: [0ο¬ β)
x y
β2 4
β1 1
0 0
1 1
2 4
vertex
40. Cubic Function f(x) = x3
β« f(x) = x3 increases on its entire domain, (ββο¬ β).
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: (ββο¬ β)
x y
β2 β8
β1 β1
0 0
1 1
2 8
41. Reciprocal Function
β« f(x) = 1/x has a vertical asymptote at x = 0
β« It is not continuous on its entire domain, (ββ, 0) ο (0ο¬ β)
Domain: (ββ, 0) ο (0ο¬ β) Range: (ββ, 0) ο (0ο¬ β)
x y
β2 β0.5
β1 β1
β0.5 β2
0.5 2
1 1
2 0.5
( )
1
f x
x
=
42. Reciprocal Squared
β« f(x) = 1/x2 has asymptotes at x = 0 and y = 0
β« It is not continuous on its entire domain, (ββ, 0) ο (0ο¬ β)
Domain: (ββ, 0) ο (0ο¬ β) Range: (0ο¬ β)
x y
β2 0.25
β1 1
β0.5 4
0.5 4
1 1
2 0.25
( ) 2
1
f x
x
=
43. Square Root Function
β« increases on its entire domain, [0ο¬ β).
β« It is continuous on its entire domain, [0ο¬ β)
Domain: [0ο¬ β) Range: [0ο¬ β)
x y
0 0
1 1
4 2
9 3
16 4
( )
f x x
=
( )
f x x
=
44. Cube Root Function
β« increases on its entire domain, (ββο¬ β).
β« It is continuous on its entire domain, (ββο¬ β)
Domain: (ββο¬ β) Range: (ββο¬ β)
x y
-8 -2
-1 -1
0 0
1 1
8 2
( ) 3
f x x
=
( ) 3
f x x
=