3-3 Writing FunctionsWriting Functions · 3-3 Writing Functions An algebraic expression that...
Transcript of 3-3 Writing FunctionsWriting Functions · 3-3 Writing Functions An algebraic expression that...
Holt McDougal Algebra 1
3-3 Writing Functions3-3 Writing Functions
Holt Algebra 1
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt McDougal Algebra 1
3-3 Writing Functions
Warm Up
Evaluate each expression for a = 2, b = –3, and c = 8.
1. a + 3c
2. ab – c
3. 12
c + b
4. 4c – b
5. ba + c
26
–14
1
35
17
Holt McDougal Algebra 1
3-3 Writing Functions
Identify independent and dependent variables.
Write an equation in function notation and evaluate a function for given input values.
Objectives
Holt McDougal Algebra 1
3-3 Writing Functions
independent variable
dependent variable
function rule
function notation
Vocabulary
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
y
5 10 15 20
1 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.
Step 3 Write an equation.
or The value of y is one-fifth of x.
The value of y is one-fifth, , of x.
15 – 4 3 and
20 – 4 4 and
10 – 4 2 and
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
y
1 2 3 4
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
y = 3x
Check It Out! Example 1 Continued
Step 2 Determine if one relationship works for the remaining values.
2 • 3 = 63 • 3 = 94 • 3 = 12
2 + 2 6 3 + 2 9 4 + 2 12
The value of y is 3 times x.
Step 3 Write an equation.
The value of y is 3 times x.
Holt McDougal Algebra 1
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 thedependent 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
Identify the independent and dependent variables
in the situation.
The height of a candle decreases d centimeters for every hour it burns.
Dependent: height of candle
Independent: time
The height of a candle depends on the number of hours it burns.
Example 2B: Identifying Independent and
Dependent Variables
Holt McDougal Algebra 1
3-3 Writing Functions
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
Identify the independent and dependent variables
in the situation.
Example 2C: Identifying Independent and
Dependent Variables
Holt McDougal Algebra 1
3-3 Writing Functions
Helpful Hint
There are several different ways to describe the variables of a function.
IndependentVariable
DependentVariable
x-values y-values
Domain Range
Input Output
x f(x)
Holt McDougal Algebra 1
3-3 Writing Functions
Check It Out! Example 2a
A company charges $10 per hour to rent a jackhammer.
Identify the independent and dependent variable in the situation.
The cost to rent a jackhammer depends onthe length of time it is rented.
Dependent variable: cost
Independent variable: time
Holt McDougal Algebra 1
3-3 Writing Functions
Identify the independent and dependent variable in the situation.
Check It Out! Example 2b
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 baby-sits. 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 yis 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 a function of x.
y = f (x)
y = f(x)
Holt McDougal Algebra 1
3-3 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 function for the amount a math tutor charges isf(h) = 35h.
Example 3A: Writing Functions
The fee a math tutor charges depends on number of hours.
Dependent: fee
Independent: hours
Let h represent the number of hours of tutoring.
Holt McDougal Algebra 1
3-3 Writing Functions
A fitness center charges a $100 initiation fee plus $40 per month.
The function for the amount the fitness center charges is f(m) = 40m + 100.
Example 3B: Writing Functions
Identify the independent and dependent variables. Write an equation in function notation for the situation.
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
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 function for the total cost of the lettuce is f(x) = 1.69x.
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.
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 function for the total park cost is
f(x) = 29.99x + 6.
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.
Holt McDougal Algebra 1
3-3 Writing Functions
You can think of a function as an input-outputmachine. For Tasha’s earnings, f(x) = 5x. If you input a value x, the output is 5x.
input
x
function
f(x)=5x
output30
2
Holt McDougal Algebra 1
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.
= 21 + 2
f(7) = 3(7) + 2 Substitute 7 for x.
f(x) = 3(x) + 2
= 23
Simplify.
f(x) = 3(x) + 2
f(–4) = 3(–4) + 2 Substitute –4 for x.
Simplify.= –12 + 2
= –10
Holt McDougal Algebra 1
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(t) = 1.5t – 5
g(6) = 1.5(6) – 5
= 9 – 5
= 4
g(–2) = 1.5(–2) – 5
= –3 – 5
= –8
Holt McDougal Algebra 1
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.
= 202 = –2
Holt McDougal Algebra 1
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(1) = 2(1) – 1
= 2 – 1
= 1
h(c) = 2c – 1
h(–3) = 2(–3) – 1
= –6 – 1
= –7
Holt McDougal Algebra 1
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.
= –5 = 101
Holt McDougal Algebra 1
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
Write a function to describe the situation. Find the reasonable domain and range of the function.
Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each, if he buys any at all.
Money spent is $15.00 for each DVD.
f(x) = $15.00 • 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.
A reasonable range for this situation is {$0, $15, $30, $45}.
x 1 2 3
f(x) 15(1) = 15 15(2) = 30 15(3) = 45
0
15(0) = 0
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 500 times the setting #.watts
f(x) = 500 • x
For each setting, the number of watts is f(x) = 500x watts.
Holt McDougal Algebra 1
3-3 Writing Functions
x
f(x)
0 1 2 3
500(0) = 0
500(1) = 500
500(2) = 1,000
500(3) =1,500
There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}.
Check It Out! Example 5
Substitute these values into the function rule to find the range values.
The reasonable range for this situation is {0, 500, 1,000, 1,500} watts.
Holt McDougal Algebra 1
3-3 Writing Functions
Lesson Quiz: Part I
Identify the independent and dependent variables. Write an equation in function notation for each situation.
1. A buffet charges $8.95 per person.
independent: number of peopledependent: costf(p) = 8.95p
2. A moving company charges $130 for weekly truck rental plus $1.50 per mile.
independent: milesdependent: cost
f(m) = 130 + 1.50m
Holt McDougal Algebra 1
3-3 Writing Functions
Lesson Quiz: Part II
Evaluate each function for the given input values.
4. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5.
f(3.5) = 20
f(–5) = –31
3. For g(t) = , find g(t) when t = 20 and
when t = –12.
g(20) = 2
g(–12) = –6
Holt McDougal Algebra 1
3-3 Writing Functions
Lesson Quiz: Part III
Write a function to describe the situation. Find the reasonable domain and range for the function.
5. A theater can be rented for exactly 2, 3, or 4 hours. The cost is a $100 deposit plus $200 per hour.
f(h) = 200h + 100
Domain: {2, 3, 4}Range: {$500, $700, $900}