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2.1 Functions and their Graphs
Function Identification
Essential Questions
How can functions be identified?
How do functions differ from relations?
How are linear functions distinguished from
nonlinear?
Relations and Functions
 A relation is a mapping, or pairing, of input
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values with output values.
The set of input values is called the domain.
The set of output values is called the range.
A relation as a function provided there is
exactly one output for each input. A vertical
line test can be used to determine if a graph is
a function.
It is NOT a function if at least one input has
more than one output.
A linear function is one that forms a line when
it is graphed (remember: linear = line). A
nonlinear function does not form a line.
Identify the Domain and Range. Then
tell if the relation is a function.
Input
Output
-3
3
1
-2
4
1
Notice the set notation!!!
Domain = {-3, 1,4}
Range = {3,-2,1,4}
4
Function?
No: input 1 is mapped onto
Both -2 & 1
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
FUNCTIONS
 Many to One Relationship
 One to One Relationship
X
1
2
3
Y
0
0
0
X
1
2
3
Y
5
7
9
Vertical Line Test
 You can use the vertical line test to visually determine if
a relation is a function.
 Slide any vertical line (pencil) across the graph to see if
any two points lie on the same vertical line.
 If there are not two points on the same vertical line then
the relation is a function.
 If there are two points on the same vertical line then the
relation is NOT a function
(b)
(a)
(c)
(d)
Graphing and Evaluating Functions
 Many functions can be represented by an equation in 2
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variables: y=2x-7
An ordered pair is a solution if the equation is true when the
values of x & y are substituted into the equation.
Ex: (2,-3) is a solution of y=2x-7 because:
-3 = 2(2) – 7
-3 = 4 – 7
-3 = -3
 In an equation, the input variable is called the independent
variable.
 The output variable is called the dependent variable and
depends on the value of the input variable.
 In y=2x-7 ….. X is the independent variable. Y is the dependant
variable.
 The graph of an equation in 2 variables is the collection of all
points (x,y) whose coordinates are solutions of the equation.
Graphing an equation in 2 variables
1. Construct a table of values
2. Graph enough solutions to
recognize a pattern
3. Connect the points with a line
or curve
Graph: y = x + 1
Step2:
Step 1
Table of values
Function Notation
 By naming the function ‘f’ you can write the function
notation:
f(x) = mx + b
 “the value of f at x”
 “f of x”
 f(x) is another name for y (grown up name)
 You can use other letters for f, like g or h
Function
Function Notation:
f(n) = n + 5
Output
Input
Function
Function Notation:
f(n) = n + 5
Rule for Function