Transcript Function Identification PowerPoint

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