FUNCTIONS & GRAPHS
2.1
JMerrill, 2006
Revised 2008
Definitions


What is domain?
Domain: the set of input values (xcoordinates)

What is range?
Range: the set of output values (ycoordinates)

Relation: a pair of quantities that are
related in some way (a set of ordered pairs)

Definitions Continued


What is a function?
A function is a dependent relationship
between a first set (domain) and a
second set (range), such that each
member of the domain corresponds to
exactly one member of the range. (i.e.
NO x-values are repeated.)
Variable Reminders




The independent/dependent variable is
the x-value
The independent/dependent variable is
the y-value
The independent variable is the
horizontal/vertical axis on an x-y plane
The dependent variable is the
horizontal/vertical axis on an x-y plane
Determine whether the following
correspondences are functions:
Numbers:
-3
9
3
2
4
YES!
Friday Night’s Date:
Juan
Boris
Nelson
Bernie
NO!
Casandra
Rebecca
Helga
Natasha
You Do: Are these
Correspondences Functions?
Numbers:
-6
36
-2
4
2
YES!
Numbers:
-3
1
5
9
2
4
6
8
NO!
Determine whether the relation is a function.
If yes, identify the domain and range

{(2,10), (3,15), (4,20)}

Yes


Domain: {2, 3, 4}. Range: {10, 15, 20}
{(-7,3), (-2,1), (-2,4), (0,7)}

No (the x-value of -2 repeats)
Determine whether the relation is a function.
If yes, identify the domain and range
Domain Range
Domain Range
-10
0
-10
0
-8
2
-8
2
-6
4
-6
4
-4
6
-4
6
-6
8
-2
8
No; -6 repeats
Yes; D:{-10, -8, -6, -4, -2};
R:{0, 2, 4, 6, 8}
Testing for Functions
Algebraically

Which of these is a function?

A. x2 + y = 1
B. -x + y2 = 1

Do you know why?

Testing for Functions
Algebraically

Which of these is a function?

A. x2 + y = 1

Solve for y: y = -x2 + 1

No matter what I substitute for x, I will
only get one y-value
Testing for Functions
Algebraically

Which of these is a function?

B. -x + y2 = 1

Solve for y:

y   1x
If x = 3 for example, y = 2 or -2. So each
x pairs with 2-different y’s. The x’s
repeat—not a function.
Function Notation



f(x) = y
So f(x) = 3x + 2 means the same thing
as y = 3x + 2
f is just the name of the function
Evaluating a Function

Let g(x) = -x2 + 4x + 1






A. Find g(2)
B. Find g(t)
C. Find g(x+2)
A. g(2) = 5
B. g(t) = -t2 + 4t + 1
C. g(x+2) = -x2 + 5
Interval Notation: Bounded Intervals

Notation Interval Type
[a,b]
Closed

(a,b)
Open
a<x<b

[a,b)
Half-open
Closed-left;
Open right
ax<b

(a,b]
Half-open
Open-left
Closed-right

Inequality
axb
a<xb
Graph
[
]
a
b
(
a
[
a
)
b
)
b
(
a
]
b
Interval Notation: Unbounded Intervals

Notation
(-,b]
Interval Type
Unbounded left
Closed
Inequality
xb

(-,b)
Unbounded left
Open
x<b

[a,)
Unbounded right
Closed
ax
[
a

(a,)
Unbounded right
Open
a<x
(
a

Graph
b
]
)
b
Domain: Graphical
[2,∞)
(-∞,∞)
Domain: Graphical
(-∞,∞)
[-3,∞)
Graphs: Are These Functions?
How
Can
You
Tell?
Yes
Yes
The
Vertical
Line
Test
No
No
Are They Functions?
Yes
No
No
Yes