Transcript document

Domain
Rule
Range
Lesson 2.3
Suppose we had a box that would
take an input, apply a rule, and spit
out the output. This would be
called a function box.
This function box
takes the name of a
movie and spits out
the first letter of
the name.
This function box
takes a number and
adds 1 to it.
This function box
takes a number
and calculates its
square root.
The numbers that go into the function
box are called the domain or the x
values. The numbers that come out of
the function box after the rule is
applied are called the
Domain
range or y values. Some
examples of rules could be
Y = 2x - 2, or y = x2.
Rule
Range
When the number going into the
function box produces a unique value
coming out of the box, we say that the
relationship between the two numbers
constitutes a “function”. This means
that you can’t put 4 into the box and
get a 6 out one time and a 10 out the
next time. The rule is often written
with “function notation”. This means
that instead of writing y = 2x - 2, the
rule is written f(x) = 2x - 2.
Definition of a Function
A function is a relationship
between two variables such
that each value of the first
variable is paired with
exactly one value of the
second variable.
How to tell if a relation between two
numbers is a function.
Ordered Pairs - A set of order pairs
is a function if every x-value produces
a unique y-value.
{(1,5), (2,9), (3,12)} This is a function
{(1,5), (2,9), (3,12), (1,7)}
This is not a function because the xvalue 1 produces both 5 and 7 y-values
How to tell if a relation between two
numbers is a function. Table: For
each x-value, there is exactly one value of y.
Do the numbers in the table represent a
function?
Domain, x
1
2
3
4
5
6
52
Range, y
-3.6
-3.6
4.2
4.2
10.7
12.1
52
This is a function
because each xvalue has only
one y-value.
How to tell if a relation between two
numbers is a function. Table: For
each x-value, there is exactly one value of y.
Do the numbers in the table represent a
function?
Domain, x
3
3
3
4
10
11
52
Range, y
7
8
10
42
34
18
52
This is not a
function because
the x-value 3 has
three different
y-values.
How to tell if a relation between
two numbers is a function.
Mapping Diagram- a mapping diagram
shows a function if each member of
the domain goes to only one member of
the range.
Range
Domain
This is a
-2
10
function
4
3
7
-7
Domain
Range
-2
4
7
10
3
-7
13
This is a not function because
the 7 in the domain produces
both a -7 and a 13 in the range
How to tell if a relation between
two numbers is a function.
Graph - a graph shows a function if it
passes the vertical line test. This
means that a vertical line drawn
anywhere on the graph will pass
through only one point on the graph.
Is this graph a function?
This parabola IS a function. We
could not draw a vertical line that
intersected the graph in more than
one place.
Is this graph a function?
This ellipse is NOT a function!
All it took was one vertical line
that intersected in more than
one place for us to be sure.
Which of these are functions?
The first two are functions. In fact, the
first graph is a linear function. There is
only one line that is NOT a function.
That's right! A vertical line is not a
function.
Weight
For the data shown, is weight a
function of height?
NO
Height
Definition of Relation
A relationship between two variables such
that each value of the first variable is
paired with one or more values of the
second variable is called a relation.
The domain is the set of all possible
values of the first variable. The
range is the set of all possible values
of the second variable.
State the domain and range:
Domain:
{-4,-1,2,6}
Range:
{-2,0,2,4}
This is a discrete function - made up
of points.
State the domain and range:
Domain:
x  -3
Range:
y  2
This is a continuous function - made up
of a line, ray, segment, or curve.
f(x) = x2
Finding the domain and
range from the function
To find the domain ask if there are any
values x cannot be. In this case the
answer is No. Therefore, the domain is
x x 
This reads x such that x is an
element of the Real Number System. To find
the range, ask what kind of numbers you will
get when you take a number and square it.
The number will always be 0 or positive.
Therefore, the range is y y  0 This reads
y such that y is greater than or equal to 0.
f(x) = x2
Finding the domain and
range from the graph
Look at the graph and ask what the x-values
are on the graph. Since the branches of the
parabola extend indefinitely, the values of x
extend to negative and positive infinity.
Therefore, the domain is x x  
Look at the graph again and
ask what the y-values are.
Notice that the lowest yvalue on the graph is 0.
Therefore the range is y y  0
f ( x) 
1
x
Finding the domain and
range from the function
Domain: Are there any values x cannot be?
x cannot be 0 because the square root of
0 = 0 and division by 0 is undefined. x
cannot be negative because we cannot take
the square root of a negative number.
Therefore the domain is xx > 0.
Range: What is y when x > 0?
y will never be 0 but can be any positive
number. Therefore the range is yy > 0.
f ( x) 
1
x
Finding the domain and
range from the graph
Find the domain.
Look at the graph The domain is xx > 0.
and ask what are
the x-values?
Find the range. Look at the
graph and ask what are the
y-values?
The range is yy > 0.
Give the domain and range of
each function.
f(x) = -x4
The domain is
The range is
x x 
yy  0
g ( x)  3 2 x
The domain is
The range is
xx > 0
yy > 0
Give the domain and
range of the function.
f(x) = x + 3
The domain is
x x 
The range is
yy  3
p. 108 (16 - 42)
Function Notation
If there is a correspondence between
values of the domain, x, and values of
the range, y, that is a function, then
y = f(x) read y = “f of x”.
(x,y) can be written (x, f(x))
The variable x is called the independent
variable and the variable y, or f(x) is called
the dependent variable.
Evaluating Functions
Let f(x) = 3x2 - 2x. Find f(4)
To evaluate a function, replace x
with the value of the number.
f(4) = 3(4)2 - 2(4) = 3(16) - 8 = 40
Let f(x) = 8x - 5. Find f(-2)
f(-2) = 8(-2) - 5 = -16 - 5 = -21
Let f(x) = 2x2 + 5x. Find f(3)
f(3) = 2(3)2 + 5(3) = 2(9) + 15 = 33
Applying Functions
Your pay (P) for working part-time at the
Aquarium is a function of the number of
hours you work (h). If you make $5.25
an hour, then P(h) = 5.25h. Find your pay
for 5 hours of work.
Since you worked 5
hours, h = 5. To find
your pay, find P(5).
P(5) = 5.25(5) = 26.25
Your pay for 5 hours is $26.25
Applying Functions
The cost (C) of tickets for the ballet is a
function of the number (n) of tickets you
buy. If each ticket costs $26.50, then
C(n) = 26.5n. Find the cost of 4 tickets.
Since you want 4 tickets
n = 4. To find the cost,
find C(4)
C(4) = 26.5(4) = 106. The
cost of the tickets is $106.
Applying Functions
Monthly residential electric charges, c, are
determined by adding a fixed fee of $6.00 to
the product of the amount of electricity
consumed each month, x, in kilowatt-hours
and a rate factor of 0.035 cents per kilowatthour. Write a linear function to model the
monthly electric charge, c, as a function of
the amount of electricity consumed
each month, x.
c = electricity used + fixed fee
c = 0.035x + 6.00
Applying Functions
If a household uses 712 kilowatt-hours of
electricity in a given month, how much is the
monthly electric charge?
Evaluate the function for x = 712
c(x) = 0.035x + 6.00
c(712) = 0.035(712) + 6.00
c(712) = 30.92
For 712 kilowatt-hours of electricity,
the monthly charge is $30.92
p. 108 (43 – 60, 64 - 69)