3 3 Writing Functions Warm Up Lesson Presentation

3 -3 Writing. Functions Warm Up Lesson Presentation Lesson Quiz Holt 1 Algebra 1 Holt. Algebra Mc. Dougal

3 -3 Writing Functions Warm Up Evaluate each expression for a = 2, b = – 3, and c = 8. 1. a + 3 c 26 2. ab – c – 14 c+b 1 3. 1 2 4. 4 c – b 35 5. ba + c 17 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Objectives Identify independent and dependent variables. Write an equation in function notation and evaluate a function for given input values. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Vocabulary independent variable function rule function notation Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 1: Using a Table to Write an Equation Determine a relationship between the x- and y -values. Write an equation. x 5 y 1 10 15 20 2 3 4 Step 1 List possible relationships between the first x or y-values. 5 – 4 = 1 or Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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 = 3 x Holt Mc. Dougal Algebra 1 The value of y is 3 times x.

3 -3 Writing Functions The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output). Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 2 A: 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 2 B: 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 2 C: 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Helpful Hint There are several different ways to describe the variables of a function. Independent Variable x-values Dependent Variable y-values Domain Range Input Output x f(x) Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Check It Out! Example 2 a 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Check It Out! Example 2 b 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 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions The dependent variable is a function of the independent variable. y is y = a function of f y = f(x) Holt Mc. Dougal Algebra 1 x. (x)

3 -3 Writing Functions Example 3 A: 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) = 35 h. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 3 B: 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) = 40 m + 100. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Check It Out! Example 3 a 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. 69 x. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Check It Out! Example 3 b 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 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. 99 x + 6. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions You can think of a function as an input-output machine. For Tasha’s earnings, f(x) = 5 x. If you input a value x, the output is 5 x. Holt Mc. Dougal Algebra 1 input x 2 6 function f(x)=5 x 10 5 x 30 output

3 -3 Writing Functions Example 4 A: Evaluating Functions Evaluate the function for the given input values. For f(x) = 3 x + 2, find f(x) when x = 7 and when x = – 4. f(x) = 3(x) + 2 f(7) = 3(7) + 2 Substitute 7 for x. = 21 + 2 Simplify. = 23 Holt Mc. Dougal Algebra 1 f(x) = 3(x) + 2 f(– 4) = 3(– 4) + 2 Substitute – 4 for x. Simplify. = – 12 + 2 = – 10

3 -3 Writing Functions Example 4 B: Evaluating Functions Evaluate the function for the given input values. For g(t) = 1. 5 t – 5, find g(t) when t = 6 and when t = – 2. g(t) = 1. 5 t – 5 g(6) = 1. 5(6) – 5 g(– 2) = 1. 5(– 2) – 5 =9– 5 = – 3 – 5 =4 = – 8 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 4 C: Evaluating Functions Evaluate the function for the given input values. For , find h(r) when r = 600 and when r = – 12. = 202 Holt Mc. Dougal Algebra 1 = – 2

3 -3 Writing Functions Check It Out! Example 4 a Evaluate the function for the given input values. For h(c) = 2 c – 1, find h(c) when c = 1 and when c = – 3. h(c) = 2 c – 1 h(1) = 2(1) – 1 h(c) = 2 c – 1 h(– 3) = 2(– 3) – 1 =2– 1 = – 6 – 1 =1 = – 7 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Check It Out! Example 4 b Evaluate the function for the given input values. For g(t) = , find g(t) when t = – 24 and when t = 400. = – 5 Holt Mc. Dougal Algebra 1 = 101

3 -3 Writing Functions 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. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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 f(x) is = $15. 00 for each • DVD. x If Joe buys x DVDs, he will spend f(x) = 15 x dollars. Joe only has enough money to purchase 1, 2, or 3 DVDs. A reasonable domain is {0, 1, 2, 3}. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Example 5 Continued Substitute the domain values into the function rule to find the range values. x 0 1 2 3 f(x) 15(0) = 0 15(1) = 15 15(2) = 30 15(3) = 45 A reasonable range for this situation is {$0, $15, $30, $45}. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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 f(x) = 500 watts 500 times • the setting #. x For each setting, the number of watts is f(x) = 500 x watts. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions 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 0 f(x) 500(0) = 0 1 500(1) = 500 2 500(2) = 1, 000 3 500(3) = 1, 500 The reasonable range for this situation is {0, 500, 1, 000, 1, 500} watts. Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Lesson Quiz: Part I Identify the independent and dependent variables. Write an equation in function notation for each situation. 1. A buffet charges $8. 95 person. independent: number of people dependent: cost f(p) = 8. 95 p 2. A moving company charges $130 for weekly truck rental plus $1. 50 per mile. independent: miles dependent: cost f(m) = 130 + 1. 50 m Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Lesson Quiz: Part II Evaluate each function for the given input values. 3. For g(t) = , find g(t) when t = 20 and when t = – 12. g(20) = 2 g(– 12) = – 6 4. For f(x) = 6 x – 1, find f(x) when x = 3. 5 and when x = – 5. f(3. 5) = 20 f(– 5) = – 31 Holt Mc. Dougal Algebra 1

3 -3 Writing Functions Lesson Quiz: Part III Write a function to describe the situation. Find the 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) = 200 h + 100 Domain: {2, 3, 4} Range: {$500, $700, $900} Holt Mc. Dougal Algebra 1