2.3 – Introduction to Functions2.3 – Introduction to Functions Objectives:Objectives:

State the domain and range of a relation, and tell State the domain and range of a relation, and tell whether it is a function.whether it is a function.

Write a function in function notation and evaluate Write a function in function notation and evaluate it.it.

Standards: Standards: 2.8.11.S. Analyze properties and relationships of 2.8.11.S. Analyze properties and relationships of

functions.functions. 2.8.11.O. Determine the domain and range of a 2.8.11.O. Determine the domain and range of a

relation.relation.

Functions and relations are commonly used to Functions and relations are commonly used to represent a variety of real-world relationships.represent a variety of real-world relationships.

A A functionfunction is a relationship between two is a relationship between two variables such that each value of the first variables such that each value of the first variable is paired with exactly one value of variable is paired with exactly one value of the second variable.the second variable.

The The domaindomain of a function is the set of all possible of a function is the set of all possible values of the first variable.values of the first variable.

The The rangerange of a function is the set of all possible of a function is the set of all possible values of the second variable.values of the second variable.

A A functionfunction may also be represented by data may also be represented by data in a table.in a table.

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

a). b).

More Examples:More Examples:xx yy

33 44

33 55

55 -4-4

66 33

Ex. 1c

NO

xx yy

22 22

44 33

66 44

88 55

Ex. 1d

YES

You can use the You can use the vertical line testvertical line test to determine if to determine if a graph represents a function.a graph represents a function.

If every vertical line intersects a given graph at no If every vertical line intersects a given graph at no more than one point, the graph represents a more than one point, the graph represents a

function.function.

a). b).

c).

Yes, it passes the vertical line test. Any vertical line drawn on this graph hits only 1 point.

d).

No, it does not pass the vertical line test. The middle segment hits the vertical line at an infinite number of points.

III. A function is a special type of III. A function is a special type of relationrelation. A relationship . A relationship between two variables such that each value of the first between two variables such that each value of the first variable is paired with one or more values of the second variable is paired with one or more values of the second variable is called a variable is called a relationrelation..

Ex 4.

D: {-4, -3, -1, 2, 3, 5}

R: {-3, 0, 1, 2}

b.

State the domain and range of State the domain and range of eacheach

graphed.graphed.c. d.

D: x ≥ -2

R: y ≥ -1

Functions and Function NotationFunctions and Function Notation yy = 2 = 2xx + 5 → + 5 → ff ( (xx) = 2) = 2xx + 5 + 5

If there is a correspondence between the domainIf there is a correspondence between the domainand range that is a function, then and range that is a function, then yy = = ff((xx), and (), and (xx, , yy) ) can be written as (can be written as (xx, , ff((xx)) The number represented by )) The number represented by ff((xx) is the value of the function ) is the value of the function f f at at xx..

The variable The variable xx is called the is called the independent independent variablevariable..

The variableThe variable y y, or, or f(x) f(x), is called the, is called the dependent dependent variablevariable..

Example 2Example 2 Monthly residential electric charges, Monthly residential electric charges, cc, are , are

determined by adding a fixed fee of $6.00 to the determined by adding a fixed fee of $6.00 to the product of the amount of electricity consumed product of the amount of electricity consumed each month, each month, x,x, in kilowatt-hours and a rate factor in kilowatt-hours and a rate factor of 0.35 cents per kilowatt-hour.of 0.35 cents per kilowatt-hour.

Write a linear function to model the monthly electric charge, Write a linear function to model the monthly electric charge, cc, as a function of the amount of electricity consumed each , as a function of the amount of electricity consumed each month, month, xx..

If a household uses 712 kilowatt-hours of electricity in a If a household uses 712 kilowatt-hours of electricity in a given month, how much is the monthly electric charge?given month, how much is the monthly electric charge?

a. C(x) = 0.35x + 6 b. C(712) = 0.35 (712) + 6 C(712) = $255.20

Example 3Example 3 A gift shop sells a specialty fruit-and-nut mix at A gift shop sells a specialty fruit-and-nut mix at

a cost of $2.99 per pound. During the holiday a cost of $2.99 per pound. During the holiday season, you can buy as much of the mix as you season, you can buy as much of the mix as you like and have it packaged in a decorative tin like and have it packaged in a decorative tin that costs $4.95.that costs $4.95. a. Write a linear function to model the total cost in a. Write a linear function to model the total cost in

dollars, c, of the tin containing the fruit-and-nut mix dollars, c, of the tin containing the fruit-and-nut mix as a function of the number of pounds of the mix, n.as a function of the number of pounds of the mix, n.

b. Find the total cost of a tin that contains 1.5 b. Find the total cost of a tin that contains 1.5 pounds of the mix.pounds of the mix.

a. C(n) = 2.99n + 4.95

b. C(1.5) = 2.99(1.5) + 4.95

C(1.5) = $9.44

Writing Activities:Writing Activities: 3a. Describe several different ways to 3a. Describe several different ways to

represent a function. Include examples.represent a function. Include examples.

3b. Give the domain and range for each 3b. Give the domain and range for each of your functions in part a.of your functions in part a.

Writing Activities:Writing Activities: 4. Give an example of a real-life 4. Give an example of a real-life

relation that is not a function.relation that is not a function.

Standard 2.8.11 O Determine the Standard 2.8.11 O Determine the domain and range of a relationdomain and range of a relation

A relation can be represented in several ways such as a set A relation can be represented in several ways such as a set of ordered pairs or a graph. How can you identify the of ordered pairs or a graph. How can you identify the domain and the range of a relation?domain and the range of a relation?

1). Consider the relation below:1). Consider the relation below: {(-2, 7), (-1,5) (-1, 4) (2,3) (3,3) (5,0)}{(-2, 7), (-1,5) (-1, 4) (2,3) (3,3) (5,0)}

2). Each graph below shows a relation. Identify the2). Each graph below shows a relation. Identify the domain and range of each relation.domain and range of each relation.