Transcript Document

3-3 Writing Functions
Objectives
Identify independent and dependent
variables.
Write an equation in function notation and
evaluate a function for given input values.
Holt McDougal Algebra 1
3-3 Writing Functions
Example 1: Using a Table to Write an Equation
Determine a relationship between the x- and
y-values. Write an equation.
x
5
y
1
10 15 20
2
3
4
Step 1 List possible relationships between the
first x or y-values.
5 – 4 = 1 or
Holt McDougal Algebra 1
3-3 Writing Functions
Example 1 Continued
Step 2 Determine if one relationship works for the
remaining values.
10 – 4  2
and
15 – 4  3
and
20 – 4  4
and
The value of y is one-fifth,
, of x.
Step 3 Write an equation.
or
The value of y is one-fifth of x.
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 1
Determine a relationship between the x- and
y-values. Write an equation.
{(1, 3), (2, 6), (3, 9), (4, 12)}
x
1
2
3
4
y
3
6
9
12
Step 1 List possible relationships between the
first x- or y-values.
1

3 = 3 or 1 + 2 = 3
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 1 Continued
Step 2 Determine if one relationship works for the
remaining values.
2 •3= 6
3 •3 = 9
4 • 3 = 12
2+26
3+29
4 + 2  12
The value of y is 3 times x.
Step 3 Write an equation.
y = 3x
Holt McDougal Algebra 1
The value of y is 3 times x.
3-3 Writing Functions
The equation in Example 1 describes a function
because for each x-value (input), there is only one
y-value (output).
Holt McDougal Algebra 1
3-3 Writing Functions
The input of a function is the independent
variable. The output of a function is the
dependent variable. The value of the
dependent variable depends on, or is a
function of, the value of the independent
variable.
Holt McDougal Algebra 1
3-3 Writing Functions
Example 2A: Identifying Independent and
Dependent Variables
Identify the independent and dependent variables
in the situation.
A painter must measure a room before
deciding how much paint to buy.
The amount of paint depends on the measurement
of a room.
Dependent: amount of paint
Independent: measurement of the room
Holt McDougal Algebra 1
3-3 Writing Functions
Example 2B: Identifying Independent and
Dependent Variables
Identify the independent and dependent variables
in the situation.
The height of a candle decreases d
centimeters for every hour it burns.
The height of a candle depends on the number of
hours it burns.
Dependent: height of candle
Independent: time
Holt McDougal Algebra 1
3-3 Writing Functions
Example 2C: Identifying Independent and
Dependent Variables
Identify the independent and dependent variables
in the situation.
A veterinarian must weigh an animal before
determining the amount of medication.
The amount of medication depends on the
weight of an animal.
Dependent: amount of medication
Independent: weight of animal
Holt McDougal Algebra 1
3-3 Writing Functions
Helpful Hint
There are several different ways to
describe the variables of a function.
Independent
Variable
x-values
Dependent
Variable
y-values
Domain
Range
Input
Output
x
f(x)
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 2a
Identify the independent and dependent
variable in the situation.
A company charges $10 per hour to rent a
jackhammer.
The cost to rent a jackhammer depends on
the length of time it is rented.
Dependent variable: cost
Independent variable: time
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 2b
Identify the independent and dependent
variable in the situation.
Apples cost $0.99 per pound.
The cost of apples depends on the number
of pounds bought.
Dependent variable: cost
Independent variable: pounds
Holt McDougal Algebra 1
3-3 Writing Functions
An algebraic expression that defines a function
is a function rule.
Suppose Tasha earns $5 for each hour she babysits. Then 5 • x is a function rule that models
her earnings.
If x is the independent variable and y is the
dependent variable, then function notation for y
is f(x), read “f of x,” where f names the function.
When an equation in two variables describes a
function, you can use function notation to write it.
Holt McDougal Algebra 1
3-3 Writing Functions
The dependent variable is a function of the
independent variable.
y
is
y
=
a function of
f
y = f(x)
Holt McDougal Algebra 1
x.
(x)
3-3 Writing Functions
Example 3A: Writing Functions
Identify the independent and dependent
variables. Write an equation in function notation
for the situation.
A math tutor charges $35 per hour.
The fee a math tutor charges depends on number
of hours.
Dependent: fee
Independent: hours
Let h represent the number of hours of tutoring.
The function for the amount a math tutor charges is
f(h) = 35h.
Holt McDougal Algebra 1
3-3 Writing Functions
Example 3B: Writing Functions
Identify the independent and dependent
variables. Write an equation in function notation
for the situation.
A fitness center charges a $100 initiation
fee plus $40 per month.
The total cost depends on the number of months,
plus $100.
Dependent: total cost
Independent: number of months
Let m represent the number of months
The function for the amount the fitness center
charges is f(m) = 40m + 100.
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 3a
Identify the independent and dependent
variables. Write an equation in function notation
for the situation.
Steven buys lettuce that costs $1.69/lb.
The total cost depends on how many pounds
of lettuce Steven buys.
Dependent: total cost
Independent: pounds
Let x represent the number of pounds Steven buys.
The function for the total cost of the lettuce is f(x) =
1.69x.
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 3b
Identify the independent and dependent
variables. Write an equation in function notation
for the situation.
An amusement park charges a $6.00 parking
fee plus $29.99 per person.
The total cost depends on the number of persons in
the car, plus $6.
Dependent: total cost
Independent: number of persons in the car
Let x represent the number of persons in the car.
The function for the total park cost is
f(x) = 29.99x + 6.
Holt McDougal Algebra 1
3-3 Writing Functions
You can think of
a function as an
input-output
machine. For
Tasha’s earnings,
f(x) = 5x. If you
input a value x,
the output is 5x.
Holt McDougal Algebra 1
input
x 2
function
f(x)=5x
30
output
3-3 Writing Functions
Example 4A: Evaluating Functions
Evaluate the function for the given input values.
For f(x) = 3x + 2, find f(x) when x = 7 and
when x = –4.
f(x) = 3(x) + 2
f(7) = 3(7) + 2 Substitute
7 for x.
f(7) = 21 + 2 Simplify.
f(7) = 23
Holt McDougal Algebra 1
f(x) = 3(x) + 2
f(–4) = 3(–4) + 2 Substitute
–4 for x.
f(–4) = –12 + 2 Simplify.
f(–4) = –10
3-3 Writing Functions
Example 4B: Evaluating Functions
Evaluate the function for the given input values.
For g(t) = 1.5t – 5, find g(t) when t = 6 and
when t = –2.
g(t) = 1.5t – 5
g(6) = 1.5(6) – 5
g(6) = 9 – 5
g(6) = 4
Holt McDougal Algebra 1
g(t) = 1.5t – 5
g(–2) = 1.5(–2) – 5
g(–2) = –3 – 5
g(–2) = –8
3-3 Writing Functions
Example 4C: Evaluating Functions
Evaluate the function for the given input values.
For
, find h(r) when r = 600
and when r = –12.
h(600)
h(600) = 202
Holt McDougal Algebra 1
h(-12)
h(-12)= –2
3-3 Writing Functions
Check It Out! Example 4a
Evaluate the function for the given input values.
For h(c) = 2c – 1, find h(c) when c = 1
and when c = –3.
h(c) = 2c – 1
h(c) = 2c – 1
h(1) = 2(1) – 1
h(–3) = 2(–3) – 1
h(1) = 2 – 1
h(–3) = –6 – 1
h(1) = 1
Holt McDougal Algebra 1
h(–3) = –7
3-3 Writing Functions
Check It Out! Example 4b
Evaluate the function for the given input values.
For g(t) =
, find g(t) when t = –24 and
when t = 400.
g(-24)
g(-24) = –5
Holt McDougal Algebra 1
g(400)
g(400) = 101
3-3 Writing Functions
When a function describes a real-world
situation, every real number is not always
reasonable for the domain and range. For
example, a number representing the
length of an object cannot be negative,
and only whole numbers can represent a
number of people.
Holt McDougal Algebra 1
3-3 Writing Functions
Example 5: Finding the Reasonable Domain and
Range of a Function
Joe has enough money to buy 1, 2, or 3 DVDs
at $15.00 each, if he buys any at all.
Write a function to describe the situation. Find
the reasonable domain and range of the function.
Money spent
f(x)
is
=
$15.00
$15.00
for each
•
DVD.
x
If Joe buys x DVDs, he will spend f(x) = 15x dollars.
Joe only has enough money to purchase 1, 2,
or 3 DVDs. A reasonable domain is {0, 1, 2, 3}.
Holt McDougal Algebra 1
3-3 Writing Functions
Example 5 Continued
Substitute the domain values into the function
rule to find the range values.
x
0
1
2
3
f(x) 15(0) = 0 15(1) = 15 15(2) = 30 15(3) = 45
A reasonable range for this situation is
{$0, $15, $30, $45}.
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 5
The settings on a space heater are the whole
numbers from 0 to 3. The total number of watts
used for each setting is 500 times the setting
number. Write a function to describe the number
of watts used for each setting. Find the
reasonable domain and range for the function.
Number of
watts used
is
f(x)
=
500
watts
500
times
•
the setting #.
x
For each setting, the number of watts is f(x) = 500x watts.
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 5
There are 4 possible settings 0, 1, 2, and 3, so a
reasonable domain would be {0, 1, 2, 3}.
Substitute these values into the function rule to find
the range values.
x
0
f(x)
500(0) =
0
1
500(1) =
500
2
500(2) =
1,000
3
500(3) =
1,500
The reasonable range for this situation is {0, 500,
1,000, 1,500} watts.
Holt McDougal Algebra 1