Writing Functions

Bellwork Evaluate each expression for a = 2, b = –3, and c = a + 3c 2. ab – c 3. 4c – b 4. b a + c 26 –

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

independent variable dependent variable function 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. Step 1 List possible relationships between the first x and y-values. Step 2 Determine which relationship works for the other x- and y- values. Step 3 Write an equation.

Try 1 Determine a relationship between the x- and y-values. Write an equation. {(1, 3), (2, 6), (3, 9), (4, 12)}

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 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

The height of a candle decreases 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 medication Independent: weight of animal Example 2C: Identifying Independent and Dependent Variables

Try 2a 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

Try 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 Independent 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)

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: charges Independent: 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 Example 3B: Writing Functions 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

Try 3a 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 Let x represent the number of pounds Steven bought.

Try 3b 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 cost Independent: 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 function f(x)=5x output 5x5x

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 spentis $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 f(x)f(x) 15(1) = 15 15(2) = 30 15(3) = 45 A reasonable range for this situation is {$15, $30, $45}.

Try 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.

There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}. Try 5