1. Objectives Identify independent and dependent variables.
Write an equation in function notation and evaluate a function for given input values.

2. Vocabulary independent variable dependent variable function rule
function notation

3. 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

4. Example 1 Continued
Step 2 Determine if one relationship works for the remaining values.
10 – 4  and
15 – 4  and
20 – 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.

5. 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 = 3

6. 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
The value of y is 3 times x.

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

8. The input of a function is the independent variable
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.

9. 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

10. 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

11. 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

12. Helpful Hint
There are several different ways to describe the variables of a function.
Independent
Variable
Dependent
Variable
x-values
y-values
Domain
Range
Input
Output x
f(x)

13. 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

14. 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

15. An algebraic expression that defines a function is a function rule
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 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.

16. The dependent variable is a function of the independent variable.
y is a function of x.
y = f (x)
y = f(x)

17. 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.

18. 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

19. 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.

20. 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.

21. You can think of a function as an input-output machine
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.
input x 2 6
function
f(x)=5x 5x 10 30
output

22. 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(x) = 3(x) + 2
Substitute
7 for x.
Substitute
–4 for x.
f(7) = 3(7) + 2
f(–4) = 3(–4) + 2 =
Simplify.
Simplify.
= –12 + 2
= 23
= –10

23. 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
g(–2) = 1.5(–2) – 5
= 9 – 5
= –3 – 5
= 4
= –8

24. 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

25. 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
= 2 – 1
= –6 – 1
= 1
= –7

26. 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

27. 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.

28. 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
is $ for each DVD.
f(x) = $ • 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}.

29. A reasonable range for this situation is {$0, $15, $30, $45}.
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
15(0) = 0
A reasonable range for this situation is {$0, $15, $30, $45}.

30. 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 times the setting #.
watts
f(x) = • x
For each setting, the number of watts is f(x) = 500x watts.

31. 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
f(x) 1 2 3
500(0) =
500(1) =
500
500(2) =
1,000
500(3) =
1,500
The reasonable range for this situation is {0, 500, 1,000, 1,500} watts.