Transcript Relations and Functions

Warm-Up for 10/14/13:
You have been given several
relations. You should decide
which relations you think
qualify as “functions” and
which ones do not. Sort them
with your partner on the
placemat. You have 5
minutes.
Relations and Functions
UNIT QUESTION: How can we use realworld situations to construct and compare
linear and exponential models and solve
problems?
Standards: MCC9-12.A.REI.10, 11, F.IF.1-7, 9, F.BF.1-3, F.LE.1-3, 5
Today’s Question:
What is a function, and how is function
notation used to evaluate functions?
Standard: MCC9-12.F.IF.1 and 2
• A relation is any set of ordered pairs.
• The domain is the set of all the first elements
of the ordered pairs. (all of the x-values)
• The range is the set of all the second
elements of the ordered pairs. (all of the yvalues)
Relations can be represented in different ways.
Ordered pairs
Table
x
y
-1
2
(–1, 2)
0
-2
(0, –2)
3
3
(3, 3)
Mapping
-1
-2
0
2
3
3
Graph
State the domain and range of each relation.
1. {(1,3),(2,4),(3,3),(4,4)}
Domain: {1, 2, 3, 4 }
Range: {3, 4 }
{(-1,2), (2,2), (2,4), (3,6)}
2.
-1
2
2
4
Domain: {-1, 2, 3 }
3
6
Range: {2, 4, 6}
3.
Domain: {1, 2, 3}
Range: {1, 2, 3}
{(1,1), (1,2), (2,1), (2,2), (3,3)}
Discrete Graph – a graph whose points _________
are not
connected.
are
Continuous Graph – a graph whose points _____
connected.
●
●
●
●
●
Continuous
_________
graph
Discrete
_________
graph
State the domain and range of the continuous graphs:
4.
D= [-2, 2]
R= [-1, 4]
5.
D: (-∞, ∞) or “All Real Numbers”
R: [-2, ∞)
Function: a relation in which each
element in the domain is paired with
exactly one element in the range.
S = {(0, 3), (-2, 5), (4, 7)} ______a
function!
is
Each x is paired with exactly one y!
is not
T = {(1, -2), (2, 3), (1, 4)} _______a
function!
The element 1 is paired with -1 and 4!
Consider each mapping. Is the relation a
function?
x
1
8
6
y
x
3
2
4
6
-9
10
y
-8
6
1
Not a function
The element 8 is paired with
3 and 2
Is a function
Consider a relation represented in tabular
form. Which relation is a function?
(a) x -2 3 4 -2
y
0 4 -1 9
Not a function
The element -2 is
paired with 0 and 9!
(b) x -1 2 5 -3
y
4 5 4 5
Is a function
Consider a relation that is represented in
graphical form: (discrete graph below!)
Suggestion: Write as a
set of ordered pairs.
T = {(1, -1), (2, 3), (1, 4)}
Not a function: The element 1 is paired with
-1 and 4.
For relations represented as graphs, try
this test:
FAIL!
Vertical Line Test: If any vertical line passes
through no more than one point of the graph
of a relation, then the relation is a function.
Determine whether each relation is a
function:
(a)
PASS!
Is a function
y
y
(b)
(c)
FAIL!
x
Not a function
PASS!
x
Is a function
Identify each graph as continuous or discrete:
Discrete
(a)
y
y
(b)
(c)
x
Continuous
x
Continuous
Equations:
Function Notation:
y = 3x + 1
f(x) = 3x + 1
y = 4 – 2x
g(x) = 4 – 2x
y = x2 + 3x – 6
h(x) = x2 + 3x – 6
The expression f(x) is read “f of x” and
represents the variable y.
f(3) represents the y value for the function, f,
when x = 3.
Closing for 10/14/13:
Go back to the Function Sorting
Activity from the Warm-Up.
Discuss with your partner if you
need to make any changes to
your original decisions on what
was a function, and why. Then
we will share.
Let f(x) = 3x +1 and h(x) = x2 + 3x – 6. Find the
value of each of the following:
(1). f(-2) = 3(-2) + 1
= -6 + 1
= -5
For the graph of f, when x = -2, y = -5.
(2). h(-3) = (-3)2 + 3(-3) – 6
=9–9–6
= -6
For the graph of h, when x = -3, y = -6.
Find the value of each expression given its graph.
3
(1). f(4) = _____
Discrete graph
(2). g(-1) = ______
-2
Continuous graph
Find the value of each expression given its graph.
2
(3). f(-2) = _____
D = {-2, -1, 1, 3, 4}
R = { -1, 0, 2, 3}
(4). g(-3) = ______
-4
D = {x| x ≥ -5}
R = { y| y ≥ -4}
For the equation y = 3x + 2 we say that
• y is written as a function of x
• y is written in terms of x.
x is called the independent variable.
y is called the dependent variable.
For the equation A = πr2 we say that
• A is written as a function of r
• A is written in terms of r.
r is called the independent variable.
A is called the dependent variable.