Transcript 2.1 Functions and their Graphs

2.1 Functions and
their Graphs
page 67
Learning Targets
• I can determine whether a given
relations is a function.
• I can represent relations and function.
• I can graph and evaluate linear
functions.
Relations
• A relation is a mapping, or pairing, of
input values with output values.
• The set of input values is called the
domain. Also called x-coordinate.
• The set of output values is called the
range. Also called y-coordinate.
• A relation as a function provided there
is exactly one output for each input.
NOTE: x values do not repeat.
• It is NOT a function if at least one
input has more than one output
Functions
• A function is a relation in which
the members of the domain (xvalues) DO NOT repeat.
• So, for every x-value there is only
one y-value that corresponds to
it.
• y-values can be repeated.
Identify the Domain and Range. Then
tell if the relation is a function.
Input (x-values)
Output (y-values)
-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 . X repeats.
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
x values do not repeat
A Relation can be represented by a
set of ordered pairs of the form (x,y)
Quadrant II
X<0, y>0
Quadrant I
X>0, y>0
Origin (0,0)
Quadrant III
X<0, y<0
Quadrant IV
X>0, y<0
Graphing Relations
• To graph the relation in
the previous example:
• Write as ordered pairs
(-3,3), (1,-2), (1,1), (4,4)
• Plot the points
(-3,3)
(4,4)
(1,1)
(1,-2)
Same with the points (-3,3),
(1,1), (3,1), (4,-2)
(-3,3)
(1,1)
(3,1)
(4,-2)
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 no 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
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
Does the graph represent a function?
Yes
x
y
Yes
x
y
Does the graph represent a function?
No
x
y
No
x
y
Does the graph represent a function?
Yes
x
y
No
x
y
Graphing and Evaluating Functions
• Many functions can be represented by an
equation in 2 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 var.
Y is the dependant var.
• 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
Step
3:
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
Decide if the function is linear. Then
evaluate for x = -2
•
•
•
•
f(x) = -x2 – 3x + 5
Not linear….
f(-2) = -(-2)2 – 3(-2) + 5
f(-2) = 7
•
•
•
•
•
•
g(x) = 2x + 6
Is linear because x is to the first power
g(-2) = 2(-2) + 6
g(-2) = 2
The domain for both is…..
All reals
Pair-Share
• Pp. 71-72 #5-48
(Even Number Only)