2.3 – Introduction to Functions

2. 2.3 – Introduction to Functions • Objectives: • State the domain and range of a relation, and tell whether it is a function. • Write a function in function notation and evaluate it. • Standards: • 2.8.11.S. Analyze properties and relationships of functions. • 2.8.11.O. Determine the domain and range of a relation.

3. Functions and relations are commonly used to represent a variety of real-world relationships. • A function is a relationship between two variables (x, y) where each value of the first variable is paired with exactly one value of the second variable. • The domain of a function is the set of all possible values of x. • The range of a function is the set of all possible values of y. • A function may also be represented by data in a table.

4. Examples: State whether the data in each table represents a function. a). b). a). b).

5. More Examples: Ex. 1d Ex. 1c NO YES

6. You can use the vertical line test to determine if a graph represents a function. • If a vertical line intersects a given graph at no more than one point, the graph represents a function. b). a).

7. c). d). Yes, it passes the vertical line test. Any vertical line drawn on this graph hits only 1 point. No, it does not pass the vertical line test. The middle segment hits the vertical line at an infinite number of points.

8. The domain of a function is the set of all possible values of x (the inputs). • The range of a function is the set of all possible values of y (the outputs). Domain: {-2, 0, 3, 8} Range: {-26, -6, 24, 74} Domain: {-6, -4, 2, 3} Range: {7, 12, 19, 39}

9. III. There is a relation between two variables when each value of the first variable is paired with one or more values of the second variable. Ex 4.

10. State the domain and range of eachfunction graphed. b.

11. Functions and Function Notation • y = 2x + 5 → f (x) = 2x + 5 An equation can represent a function. In this case, then y = f(x), and (x, y) can be written as (x, f(x)) The number represented by f(x) is the value of the function f at x. • The variable x is called theindependent variable. • The variable y, or f(x), is called thedependent variable.

14. Example • Monthly residential electric charges, C, are determined by adding a fixed fee of $6.00 to the product of the amount of electricity consumed each month, x, in kilowatt-hours and a rate factor of 0.35 cents per kilowatt-hour. • Write a linear function to model the monthly electric charge, C, as a function of the amount of electricity consumed each month, x. • If a household uses 712 kilowatt-hours of electricity in a given month, how much is the monthly electric charge? b. C(712) = 0.35 (712) + 6 C(712) = $255.20 • C(x) = 0.35x + 6

15. Example • A gift shop sells a specialty fruit-and-nut mix at a cost of $2.99 per pound. During the holiday season, you can buy as much of the mix as you like and have it packaged in a decorative tin that costs $4.95. • a. Write a linear function to model the total cost in dollars, C, of the tin containing the fruit-and-nut mix as a function of the number of pounds of the mix, n. • b. Find the total cost of a tin that contains 1.5 pounds of the mix. • C(n) = 2.99n + 4.95 • C(1.5) = 2.99(1.5) + 4.95 • C(1.5) = $9.44

16. Writing Activities: • 3a. Describe several different ways to represent a function. Include examples. • 3b. Give the domain and range for each of your functions in part a.

17. Writing Activities: • 4. Give an example of a real-life relation that is not a function.

18. Standard 2.8.11.O: Determine the domain and range of a relation A relation can be represented in several ways such as a set of ordered pairs or a graph. How can you identify the domain and the range of a relation? 1). Consider the relation below: {(-2, 7), (-1,5) (-1, 4) (2,3) (3,3) (5,0)} 2). Each graph below shows a relation. Identify the domain and range of each relation.

20. Homework Integrated Algebra II- Section 2.3 Level A Honors Algebra II- Section 2.3 Level B