Chapter 3 3-3 Writing Functions

SAT Problem of the day Ben travels a certain distance at 25 miles per hour and returns across the same distance at 50 miles per hour. What is the average speed in miles per hour for the round trip? A)37.5 B)33.3 C)32 D)29.5 E)can not be determined from the given information

Solution to the SAT problem Right Answer: B

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

Example#1 Determine a relationship between the x- and y-values. Write an equation. Solution: Step 1 List possible relationships between the first x or y-values. x y 5 10 15 20 1 2 3 4

Solution Step 2 Determine if one relationship works for the remaining values. Step 3 Write an equation or

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

solution Step Write an equation. y = 3x

Student guided practice Do problems 3 and 4 in your book page 183

What is a function? A function is a relation where no x’s repeat. It can be in a table, graph, equation. A function is also a relation where there is an input and an output.

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

Function What is a independent variable? The input of a function is the independent variable. What is an dependent 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#3 Identify the independent and dependent variables in the situation. A painter must measure a room before deciding how much paint to buy. Solution: The amount of paint depends on the measurement of a room. Dependent: amount of paint Independent: measurement of the room

Example#4 Identify the independent and dependent variables in the situation. The height of a candle decreases d centimeters for every hour it burns. Solution: The height of a candle depends on the number of hours it burns. Dependent: height of candle Independent: time

Example#5 Identify the independent and dependent variables in the situation. A veterinarian must weigh an animal before determining the amount of medication. Solution: 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 Dependent x-values y-values Domain Range Input Output x f(x)

Student guided practice Do problems 5 and 6 in your book page 183

Function rule and function notation 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.

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

Example#6 Identify the independent and dependent variables. Write an equation in function notation for the situation. A math tutor charges $35 per hour. Solution: The fee a math tutor charges depends on number of hours. Dependent: fee Independent: hours Let h represent the number of hours of tutoring and f(h) the cost f(h) = 35h.

Example#7 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. Solution: The total cost depends on the number of months, plus $100. Dependent: total cost Independent: number of months f(m) = 40m + 100.

Student guided practice Do problems 7 and 8 in your book page 183

functions 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.

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.

example Evaluate the function for the given input values. For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.

Example Evaluate the function for the given input values. For , find h(r) when r = 600 and when r = –12.

Student guided practice Do problems 9-11 in your book page 183

Applications 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 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. f(x) = $15.00 • x 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}.

Homework!!! Do problems 13-19 in your book page 183

Closure Today we learned about writing functions Next class we are going to learned about graphing functions.