UNIT 4.3/4.4 WRITING FUNCTIONSUNIT 4.3/4.4 WRITING FUNCTIONS

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

Identify independent and dependent variables.

Write an equation in function notation and evaluate a function for given input values.

Objectives

independent variabledependent variablefunction rule function notation

Vocabulary

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

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.

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

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.

The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).

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.

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

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

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

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)

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

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

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.

The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

y = f(x)

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.

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

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.

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.

You can think of a function as an input-output machine.

input

10

x

functionf(x)=5x

output

5x

6

30

2

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

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

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

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

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

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.

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}.

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}.

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.

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.

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

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

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}

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