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

Vocabulary independent variable dependent variable function rule function notation

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

Example 1 Continued Step 2 Determine if one relationship works for the remaining 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 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 1 + 2 = 3

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.

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 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 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

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

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

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)

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

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

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.

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

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.

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 + 100.

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.

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.

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

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 = 21 + 2 Simplify. Simplify. = –12 + 2 = 23 = –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 g(–2) = 1.5(–2) – 5 = 9 – 5 = –3 – 5 = 4 = –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(c) = 2c – 1 h(1) = 2(1) – 1 h(–3) = 2(–3) – 1 = 2 – 1 = –6 – 1 = 1 = –7

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

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 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 $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}.

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}.

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.

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.