3-3 Writing Functions

Transcription

1 Example 1: Using a Table to Write an Equation Determine a relationship between the x- and y-values. Write an equation. x y Step 1 List possible relationships between the first x or y-values. 5 4 = 1 or

2 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 y

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

4 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.

5 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

6 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

7 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

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

9 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

10 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

11 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.

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

13 Identify the independent and dependent variables. Write an equation 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 fee a math tutor charges depends on number of hours. Dependent: fee Independent: hours Let h represent the number of hours of tutoring.

14 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 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

15 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.

16 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.

17 You can think of a function as an input-output machine. For Tosha s earnings, f(x) = 5x. If you input a value x, the output is 5x. input x 2 function f(x)=5x 30 output

18 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(7)= {23} = f(-4)={ 10}

19 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(6)= {4} g(t)= { 8}

20 Example 4C: Evaluating Functions Evaluate the function for the given input values. For, find h(r) when r = 600 and when r = 12. h(r)= {202} h(r)= { 2}

21 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(1)= {1} h(-3)= { 7}

22 Check It Out! Example 4b Evaluate the function for the given input values. For g(t) = when t = 400., find g(t) when t = 24 and g(-24)= { 5} g(400)= {101}

23 If f(x)=6x+1 and g(x)=-2x-7, find h(x)=f(x)+g(x) h(x)= 4x-6

24 Find the value of x so that the function has the given value. 1.h(x)=13x-4; f(x)=-4x+3; 18

25 Find the inverse function of f(x)=3x+7

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28 Warm-up Simplify ( 64) In your notes, find the inverse of the function. f x = 1 x 7 3

29 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.

30 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. 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}.

31 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. For each setting, the number of watts is f(x) = 500x watts. There are 4 possible settings 0, 1, 2, and 3, so a reasonable domain would be {0, 1, 2, 3}. The reasonable range for this situation is {0, 500, 1,000, 1,500} watts.

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33 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 per person. independent: number of people dependent: cost f(p) = 8.95p 2. A moving company charges $130 for weekly truck rental plus $1.50 per mile. independent: miles dependent: cost f(m) = m

34 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) = 6x 1, find f(x) when x = 3.5 and when x = 5. f(3.5) = {20} f( 5) = { 31}

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36 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) = 200h Domain: {0, 2, 3, 4} Range: {$0, $500, $700, $900}

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50 Class Work/ Homework Page 132, 133, 136