algebra unit 4.3.4.4.2
DESCRIPTION
Unit 4.3/4.4TRANSCRIPT
![Page 1: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/1.jpg)
UNIT 4.3/4.4 WRITING FUNCTIONSUNIT 4.3/4.4 WRITING FUNCTIONS
![Page 2: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/2.jpg)
Warm UpEvaluate 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
![Page 3: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/3.jpg)
Identify independent and dependent variables.
Write an equation in function notation and evaluate a function for given input values.
Objectives
![Page 4: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/4.jpg)
independent variabledependent variablefunction rule function notation
Vocabulary
![Page 5: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/5.jpg)
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 and y-values.
5 – 4 = 1 and
![Page 6: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/6.jpg)
Example 1 ContinuedStep 2 Determine which relationship works for the other x- and y- 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.
![Page 7: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/7.jpg)
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- and y-values.
1 • 3 = 3 and 1 + 2 = 3
![Page 8: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/8.jpg)
y = 3x
Check It Out! Example 1 Continued
Step 2 Determine which relationship works for the other x- and y- 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.
![Page 9: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/9.jpg)
The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).
![Page 10: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/10.jpg)
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.
![Page 11: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/11.jpg)
Example 2A: Identifying Independent and Dependent Variables
Identify the independent and dependent variablesin 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 paintIndependent: measurement of the room
![Page 12: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/12.jpg)
Identify the independent and dependent variablesin the situation.
The height of a candle decrease 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
![Page 13: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/13.jpg)
A veterinarian must weight an animal before determining the amount of medication.
The amount of medication depends on the weight of an animal.
Dependent: amount of medicationIndependent: weight of animal
Identify the independent and dependent variablesin the situation.
Example 2C: Identifying Independent and Dependent Variables
![Page 14: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/14.jpg)
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)
![Page 15: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/15.jpg)
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 on the length of time it is rented.
Dependent variable: costIndependent variable: time
![Page 16: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/16.jpg)
Identify the independent and dependent variable in the situation.
Check It Out! Example 2b
Camryn buys p pounds of apples at $0.99 per pound.
The cost of apples depends on the number of pounds bought.
Dependent variable: costIndependent variable: pounds
![Page 17: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/17.jpg)
An algebraic expression that defines a function is a function rule.
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.
![Page 18: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/18.jpg)
The dependent variable is a function of the independent variable.
y is a function of x.
y = f (x)
y = f(x)
![Page 19: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/19.jpg)
Identify the independent and dependent variables. Write a rule in function notation for the situation.
A math tutor charges $35 per hour.
The function for the amount a math tutor charges is f(h) = 35h.
Example 3A: Writing Functions
The amount a math tutor charges depends on number of hours.
Dependent: chargesIndependent: hours
Let h represent the number of hours of tutoring.
![Page 20: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/20.jpg)
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 FunctionsIdentify the independent and dependent variables. Write a rule in function notation for the situation.
The total cost depends on the number of months, plus $100.
Dependent: total costIndependent: number of months
Let m represent the number of months
![Page 21: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/21.jpg)
Check It Out! Example 3aIdentify the independent and dependent variables. Write a rule in function notation for the situation.
Steven buys lettuce that costs $1.69/lb.
The function for cost of the lettuce is f(x) = 1.69x.
The total cost depends on how many pounds of lettuce that Steven buys.
Dependent: total costIndependent: pounds
Let x represent the number of pounds Steven bought.
![Page 22: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/22.jpg)
Check It Out! Example 3bIdentify the independent and dependent variables. Write a rule 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 costIndependent: number of persons in the car
Let x represent the number of persons in the car.
![Page 23: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/23.jpg)
You can think of a function as an input-output machine.
input
10
x
functionf(x)=5x
output
5x
6
30
2
![Page 24: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/24.jpg)
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
![Page 25: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/25.jpg)
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
![Page 26: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/26.jpg)
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
![Page 27: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/27.jpg)
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
![Page 28: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/28.jpg)
Check It Out! Example 4b
Evaluate each function for the given input values.
For g(t) = , find g(t) when t = –24 and
when t = 400.
= –5 = 101
![Page 29: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/29.jpg)
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.
![Page 30: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/30.jpg)
Example 5: Finding the Reasonable Range and Domain of a Function
Write a function to describe the situation. Find a reasonable domain and range of the function.
Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each.
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 {1, 2, 3}.
![Page 31: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/31.jpg)
Example 5 Continued
Substitute the domain values into the function rule to find the range values.
x 1 2 3
f(x) 15(1) = 15 15(2) = 30 15(3) = 45
A reasonable range for this situation is {$15, $30, $45}.
![Page 32: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/32.jpg)
Check It Out! Example 5
The settings on a space heater are the whole numbers from 0 to 3. The total of watts used for each setting is 500 times the setting number. Write a function rule to describe the number of watts used for each setting. Find a 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.
![Page 33: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/33.jpg)
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.
A reasonable range for this situation is {0, 500, 1,000, 1,500} watts.
![Page 34: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/34.jpg)
Lesson Quiz: Part I
Identify the independent and dependent variables. Write a rule 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: costf(m) = 130 + 1.50m
![Page 35: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/35.jpg)
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) = 20f(–5) = –31
3. For g(t) = , find g(t) when t = 20 and
when t = –12.
g(20) = 2g(–12) = –6
![Page 36: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/36.jpg)
Lesson Quiz: Part III
Write a function to describe the situation. Find a 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 + 100Domain: {2, 3, 4}Range: {$500, $700, $900}
![Page 37: Algebra unit 4.3.4.4.2](https://reader036.vdocument.in/reader036/viewer/2022062308/559c44211a28abdd0b8b4619/html5/thumbnails/37.jpg)
All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.