Transcript 1-7 - Cloudfront.net

1-7
1-7 Function
Function Notation
Notation
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
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Function Notation
Warm Up
Evaluate.
1. 5x – 2 when x = 4
18
2. 3x2 + 4x – 1 when x = 5 94
3.
when x = 16
48
4. 2 – t2 when
5. Give the domain and range for this
relation: {(1, 1), (–1, 1), (2, 4), (–2, 4),
(–3, 9), (3, 9)}.
D: {–3, –2, –1, 1, 2, 3}
Holt Algebra 2
R: {1, 4, 9}
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Function Notation
Objectives
Write functions using function notation.
Evaluate and graph functions.
Holt Algebra 2
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Function Notation
Vocabulary
function notation
dependent variable
independent variable
Holt Algebra 2
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Function Notation
Some sets of ordered pairs can be described by
using an equation. When the set of ordered
pairs described by an equation satisfies the
definition of a function, the equation can be
written in function notation.
Holt Algebra 2
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Function Notation
Output value
Input value
ƒ(x) = 5x + 3
ƒ of x equals 5 times x plus 3.
Holt Algebra 2
Output value
Input value
ƒ(1) = 5(1) + 3
ƒ of 1 equals 5 times 1 plus 3.
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Function Notation
The function described by ƒ(x) = 5x + 3 is the same
as the function described by y = 5x + 3. And both of
these functions are the same as the set of ordered
pairs (x, 5x+ 3).
y = 5x + 3
(x, y)
ƒ(x) = 5x + 3
(x, ƒ(x))
(x, 5x + 3)
Notice that y = ƒ(x)
(x, 5x + 3) for each x.
The graph of a function is a picture of the
function’s ordered pairs.
Holt Algebra 2
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Function Notation
Caution
f(x) is not “f times x” or “f multiplied by x.” f(x)
means “the value of f at x.” So f(1) represents
the value of f at x =1
Holt Algebra 2
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Function Notation
Example 1A: Evaluating Functions
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = 8 + 4x
Substitute each value for x and evaluate.
ƒ(0) = 8 + 4(0) = 8
ƒ
=8+4
= 10
ƒ(–2) = 8 + 4(–2) = 0
Holt Algebra 2
, and
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Function Notation
Example 1B: Evaluating Functions
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
Use the graph to find the
corresponding y-value for
each x-value.
ƒ(0) = 3
ƒ
=0
ƒ(–2) = 4
Holt Algebra 2
, and
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Function Notation
Check It Out! Example 1a
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = x2 – 4x
Holt Algebra 2
, and
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Function Notation
Check It Out! Example 1b
For each function, evaluate ƒ(0), ƒ
ƒ(–2).
ƒ(x) = –2x + 1
Holt Algebra 2
, and
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Function Notation
In the notation ƒ(x), ƒ is the name of the function.
The output ƒ(x) of a function is called the
dependent variable because it depends on the
input value of the function. The input x is called the
independent variable. When a function is
graphed, the independent variable is graphed on
the horizontal axis and the dependent variable is
graphed on the vertical axis.
Holt Algebra 2
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Function Notation
Holt Algebra 2
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Function Notation
Example 2A: Graphing Functions
Graph the function.
{(0, 4), (1, 5), (2, 6), (3, 7), (4, 8)}
Graph the points.
Do not connect the
points because the
values between the
given points have
not been defined.
Holt Algebra 2
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Function Notation
Reading Math
A function whose graph is made up of
unconnected points is called a discrete function.
Holt Algebra 2
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Function Notation
Example 2B: Graphing Functions
Graph the function f(x) = 3x – 1.
Make a table.
Graph the points.
x
3x – 1
f(x)
–1
3(– 1) – 1
–4
0
3(0) – 1
–1
1
3(1) – 1
2
Connect the points with a line because
the function is defined for all real numbers.
Holt Algebra 2
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Function Notation
Check It Out! Example 2a
Graph the function.
3 5 7 9
2 6 10
Graph the points.
Do not connect the
points because the
values between the
given points have not
been defined.
Holt Algebra 2
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Function Notation
Check It Out! Example 2b
Graph the function f(x) = 2x + 1.
Graph the points.
Make a table.
x
2x + 1
f(x)
–1
2(– 1) + 1
–1
0
2(0) + 1
1
1
2(1) + 1
3
Connect the points with a line because
the function is defined for all real numbers.
Holt Algebra 2
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Function Notation
My little sister, Savannah, is three years
old. She has a piggy bank that she wants
to fill. She started with five pennies and
each day when I come home from school,
she is excited when I give her three
pennies that are left over from my lunch
money. Create a mathematical model for
the number of pennies in the piggy bank
on day n.
Holt Algebra 2
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Function Notation
I’m more sophisticated than my
little sister so I save my money in a
bank account that pays me
3% interest on the money in the
account at the end of each month.
(If I take my money out
before the end of the month, I don’t
earn any interest for the month.) I
started the account with
$50 that I got for my birthday.
Create a mathematical model of the
amount of money I will have
in the account after m months.
Holt Algebra 2
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Function Notation
The algebraic expression used to define
a function is called the function rule. The
function described by f(x) = 5x + 3 is
defined by the function rule 5x + 3. To
write a function rule, first identify the
independent and dependent variables.
Holt Algebra 2
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Function Notation
Example 3A: Entertainment Application
A carnival charges a $5 entrance fee and $2
per ride.
Write a function to represent the total cost
after taking a certain number of rides.
Let r be the number of rides and let C be the total cost
in dollars. The entrance fee is constant.
First, identify the independent and dependent variables.
Cost depends on the entrance fee plus the number of rides taken
Dependent variable
Independent variable
Cost = entrance fee + number of rides taken
C(r) = 5 + 2r
Holt Algebra 2
Replace the words with expressions.
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Function Notation
Example 3B: Entertainment Application
A carnival charges a $5 entrance fee and $2
per ride.
What is the value of the function for an input
of 12, and what does it represent?
C(12) = 5 + 2(12)
Substitute 12 for r and simplify.
C(12) = 29
The value of the function for an input of 12 is 29. This
means that it costs $29 to enter the carnival and take 12
rides.
Holt Algebra 2
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Function Notation
Check It Out! Example 3a
A local photo shop will develop and print the
photos from a disposable camera for $0.27 per
print.
Write a function to represent the cost of photo
processing.
Let x be the number of photos and let f be the total cost
of the photo processing in dollars.
First, identify the independent and dependent variables.
Cost depends on the number of photos processed
Dependent variable
Independent variable
Cost = 0.27  number of photos processed
f(x) = 0.27x
Holt Algebra 2
Replace the words with expressions.
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Function Notation
Check It Out! Example 3b
A local photo shop will develop and print the
photos from a disposable camera for $0.27 per
print.
What is the value of the function for an input
of 24, and what does it represent?
f(24) = 0.27(24)
Substitute 24 of x and simplify.
= 6.48
The value of the function for an input of 24 is 6.48.
This means that it costs $6.48 to develop 24 photos.
Holt Algebra 2
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Function Notation
Our family has a small pool for relaxing in the
summer that holds 1500 gallons of water. I
decided to fill the pool for the summer. When I
had 5 gallons of water in the pool, I decided
that I didn’t want to stand outside and watch
the pool fill, so I had to figure out how long it
would take so that I could leave, but come
back to turn off the water at the right time. I
checked the flow on the hose and found that
it was filling the pool at a rate of 2 gallons
every minute. Create a mathematical model
for the number of gallons of water in the pool
at t minutes.
Holt Algebra 2
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Function Notation
At the end of the summer, I decide to
drain the swimming pool. I noticed that
it drains faster when there is more
water in the pool. That was interesting
to me, so I decided to measure the
rate at which it drains. I found that it
was draining at a rate of 3% every
minute. Create a mathematical model
of the gallons of water in the pool at t
minutes.
Holt Algebra 2
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Function Notation
Lesson Quiz: Part I
For each function, evaluate
1. f(x) = 9 – 6x 9; 6; 21
2.
4; 6; 0
3. Graph f(x)= 4x + 2.
Holt Algebra 2
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Function Notation
Lesson Quiz: Part II
4. A painter charges $200 plus $25 per can of paint
used.
a. Write a function to represent the total charge
for a certain number of cans of paint.
t(c) = 200 + 25c
b. What is the value of the function for an input
of 4, and what does it represent?
300; total charge in dollars if 4 cans of
paint are used.
Holt Algebra 2