Warm Up Identify the independent and dependent variables in each situation. 1. In the winter, more electricity is used when the temperature goes down, and less is used when the temperature rises. Evaluate each function for the given input values. 2. For f(x) = 5x, find f(x) when x=6 and when x=7.5 D: amount of electricity I: temperature 30 and 37.5

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

Directions: Determine a relationship between the x- and y-values. Write an equation.

Example 1 x y 5 10 15 20 1 2 3 4 Step 1 List possible relationships between the first x and y-values. 5 – 4 = 1 and

Example 1 Continued Step 2 Determine which relationship works for the other x- and y- 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.

Example 2 {(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- and y-values. 1  3 = 3 and 1 + 2 = 3

Example 2 Continued Step 2 Determine which relationship works for the other x- and y- 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).

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) y = f(x)

Directions: Identify the independent and dependent variables. Write a rule in function notation for the situation.

Example 3 A math tutor charges $35 per hour. The amount a math tutor charges depends on number of hours. Dependent: charges 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 4 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.

Example 5 Steven buys lettuce that costs $1.69/lb. 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. The function for cost of the lettuce is f(x) = 1.69x.

Example 6 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.

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.

Directions: Write a function to describe the situation. Find a reasonable domain and range of the function.

Example 7 Joe has enough money to buy 1, 2, or 3 DVDs at $15.00 each. 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 {1, 2, 3}.

A reasonable range for this situation is {$15, $30, $45}. Example 7 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 A reasonable range for this situation is {$15, $30, $45}.

Example 8 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. 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.

A reasonable range for this situation is {0, 500, 1,000, 1,500} watts. Example 8 Continued 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 A reasonable range for this situation is {0, 500, 1,000, 1,500} watts.

Lesson Summary: Part I Write a rule 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) = 130 + 1.50m

Lesson Summary: Part II Write a function to describe the situation. Find a 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 + 100 Domain: {2, 3, 4} Range: {$500, $700, $900}