Vocabulary
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Dependent Variable
Independent Variable
Input
Output
Function
Linear Function
Definition
• Dependent Variable – A variable whose value
depends on some other value.
– Generally, y is used for the dependent variable.
• Independent Variable – A variable that doesn’t
depend on any other value.
– Generally, x is used for the independent variable.
• The value of the dependent variable depends on
the value of the independent variable.
Independent and
Dependent Variables
On a graph;
the independent
variable is on
the horizontal
or x-axis.
the dependent
variable is on
the vertical
or y-axis.
y
dependent
x
independent
Example:
Identify the independent and dependent variables
in the situation.
A veterinarian must weight an animal before determining the
amount of medication.
The amount of medication depends on the weight of an
animal.
Dependent: amount of medication
Independent: weight of animal
Your Turn:
Identify the independent and dependent variable in the
situation.
A company charges $10 per hour to rent a jackhammer.
The cost to rent a jackhammer depends on the length of
time it is rented.
Dependent variable: cost
Independent variable: time
Your Turn:
Identify the independent and dependent variable in the
situation.
Camryn buys p pounds of apples at $0.99 per pound.
The cost of apples depends on the number of pounds
bought.
Dependent variable: cost
Independent variable: pounds
Definition
• Input – Values of the independent variable.
– x – values
– The input is the value substituted into an equation.
• Output – Values of the dependent variable.
– y – values.
– The output is the result of that substitution in an
equation.
Function
• In the last 2 problems you can describe the
relationship by saying that the perimeter
(dependent variable – y value) is a function
of the number of figures (independent
variable – x value).
• A function is a relationship that pairs each
input value with exactly one output value.
Function
You can think of a function as
an input-output machine.
input
x2
function
y = 5x
30
output
Helpful Hint
There are several different ways to describe the
variables of a function.
Independent
Variable
Dependent
Variable
x-values
y-values
Input
Output
Domain
Range
x
f(x)
A function is a set of ordered pairs (x, y) so that
each x-value corresponds to exactly one y-value.
Function Rule
Output
variable
Input
variable
Some functions can be described by a rule written in words, such
as “double a number and then add nine to the result,” or by an
equation with two variables. One variable (x) represents the input,
and the other variable (y) represents the output.
Linear Function
• Another method of representing a function is with
a graph.
• A linear function is a function whose graph is a
nonvertical line or part of a nonvertical line.
Example: Representing a Linear Function
A DVD buyers club charges a $20 membership fee and
$15 per DVD purchased. The table below represents
this situation.
Number of DVDs
purchased
x
0
1
2
3
4
5
Total cost ($)
y
20
35
50
65
80
95
+15
+15
+15
+15
+15
Find the first differences for the total cost.
constant
linear
Since the data shows a ___________
difference the pattern is __________.
If a pattern is linear then its graph is a straight _________.
line
Relations
A
relation is a mapping, or pairing, of
input values with output values.
 The
set of input values is called the
domain.
 The
set of output values is called the
range.
Domain & Range
Domain is the set of
all x values.
Range is the set of all
y values.
Example 1: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
Domain- D: {1, 2}
Range- R: {1, 2, 3}
Example 2:
Find the Domain and Range of the
following relation:
{(a,1), (b,2), (c,3), (e,2)}
Domain: {a, b, c, e}
Range: {1, 2, 3}
Page 107
(points
Every equation has solution points
which satisfy the equation).
3x + y = 5
Some solution points:
(0, 5), (1, 2), (2, -1), (3, -4)
Most equations have infinitely
many solution points.
Page 111
Ex 3. Determine whether the given ordered
pairs are solutions of this equation.
(-1, -4) and (7, 5); y = 3x -1
The collection of all solution points is
the graph of the equation.
3.3 Functions
•A relation as a function provided there is
exactly one output for each input.
•It is NOT a function if at least one input has
more than one output
Page 116
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE
OUTPUT
INPUT
Functions
(DOMAIN)
FUNCTION
MACHINE
OUTPUT (RANGE)
Example 6
Which of the following relations are
functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the
same first coordinate
(and different second coordinates).
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
1
3
-2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}
The Vertical Line Test
If it is possible for a vertical line
to intersect a graph at more
than one point, then the graph
is NOT the graph of a function.
Page 117
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(4,4)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
Examples
 I’m
going to show you a series of
graphs.
 Determine whether or not these
graphs are functions.
 You do not need to draw the graphs in
your notes.
#1
Function?
#2 Function?
#3 Function?
#4 Function?
#5
Function?
#6
Function?
#7 Function?
#8 Function?
#9 Function?
#10
Function?
#11
Function?
#12
Function?
Function Notation
f (x )
“f of x”
Input = x
Output = f(x) = y
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x
x
y
x
f(x)
-2
12
-2
12
-1
9
-1
9
0
6
0
6
1
3
1
3
2
0
2
0
(x, y)
(input, output)
(x, f(x))