Ind-Dep Variables and Functions

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Warm Up

Lesson Presentation

Lesson Quiz

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Warm Up

Evaluate each expression for a = 2, b = –3, and c = 8.

1. a + 3c

2. ab – c

3.

1

2 c + b

4. 4c – b

5. ba + c

26

–14

1

35

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Identify independent and dependent

variables.

Write an equation in function notation and

evaluate a function for given input values.

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independent variable

dependent variable

function rule

function notation

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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

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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.

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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 medication Independent: weight of animal

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Helpful Hint

There are several different ways to describe the variables of a function.

Independent

Variable DependentVariable x-values y-values

Domain Range

Input Output

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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.

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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: cost

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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

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The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

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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 Example 3A: Writing Functions

The amount a math tutor charges depends on number of hours.

Dependent: charges Independent: hours

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A fitness center charges a $100 initiation fee plus $40 per month.

Example 3B: Writing Functions Identify the independent and dependent

The total cost depends on the number of months, plus $100.

Dependent: total cost

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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 cost Independent: pounds

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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

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You can think of

a function as an

input-output

machine.

input

10 x

function f(x)=5x

output 5_x 6

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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.

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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

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Example 4C: Evaluating Functions

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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

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Evaluate each function for the given input values.

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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

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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

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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

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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

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x f(x)

0 1 2 3

500(0) =

0 500500(1) = 5001,000(2) = 5001,500(3) = 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.

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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 people dependent: cost

f(p) = 8.95p

2. A moving company charges $130 for weekly truck rental plus $1.50 per mile.

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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

3. For g(t) = , find g(t) when t = 20 and when t = –12.

g(20) = 2

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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 + 100 Domain: {2, 3, 4}