1.6 Relations and Functions

Identify the domain and range of relations and functions.

Determine whether a relation is a function.

Objectives

Vocabulary

relationdomainrangefunction

relation • pairing of input values with output values• set of ordered pairs (x,y), where x is an input and y is an output

domain•set of input values for a relation

range•set of output values

A

B

C

2

Domain Range

Mapping Diagram

Set of Ordered Pairs

{(2, A), (2, B), (2, C)}

(x, y) (input, output) (domain, range)

Give the domain and range for each relation:

{(100,5), (120,5), (140,6), (160,6), (180,12)}

Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C.

Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range.

Determine whether each relation is a function.

Give the domain and range.A. from the items in a store to their prices on a

certain date

B. from types of fruits to their colors

C.

D. from the number of items in a grocery cart to the total cost of the items in the cart

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.

1.7 Function Notation

Write functions using function notation.

Evaluate and graph functions.

Objectives

Vocabulary

function notationdependent variableindependent variable

ƒ(x) = 5x + 3 ƒ(1) = 5(1) + 3

Output value Output value Input valueInput value

ƒ of x equals 5 times x plus 3. ƒ of 1 equals 5 times 1 plus 3.

•f(x) is not “f times x” or

“f multiplied by x.”

•f(x) is pronounced “f of x”

•f(x) means “the value of the function at x.”

So f(1) represents the value of y at x =1

To find the range given the domain

Substitute each value of the domain intothe function and solve for f(x).

Example

ƒ(x) = 8 + 4x

For each function, evaluate ƒ(0), ƒ , and ƒ(–2).

Example

For each function, evaluate ƒ(0), ƒ , and ƒ(–2).

ƒ(x) = x2 – 4x

For each function, evaluate ƒ(0), ƒ , and ƒ(–2).

Use the graph to find the corresponding y-value for each x-value.

To find the domain given the range

Substitute each value of the range into f(x).You then have an equation.Solve for x.

ƒ(x) = 8 + 4x

EX: Find the domain given the range, y=16

In the notation ƒ(x), ƒ is the name of the function. The output ƒ(x) of a function is called the dependent variable because it depends on the input value of the function.

The input x is called the independent variable.

When a function is graphed, the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

DISCRETE Functions

{(0, 4), (1, 5), (2, 6), (3, 7), (4, 8)}

Graph the points.

Do not connect the points because the values between the given points have not been defined.

A function whose graph is made up of unconnected points is called a discrete function.

What is the domain and range ofthis discrete function?

A function whose graph is made up of connected points is called a continuous function.

Domains and ranges of continuousfunctions are inequality statementsor R.

The algebraic expression used to define a function is called the function rule.

The function described by f(x) = 5x + 3 is defined by the function rule 5x + 3.

To write a function rule, first identify the independent and dependent variables.

Example

A carnival charges a $5 entrance fee and $2 per ride.

Write a function to represent the total cost after taking a certain number of rides.

Let r be the number of rides and let C be the total cost in dollars. The entrance fee is constant.

C(r) = 5 + 2r

First, identify the independent and dependent variables.

Cost depends on the entrance fee plus the number of rides taken

Cost = entrance fee + number of rides taken

Replace the words with expressions.

Dependent variable Independent variable

What is the value of the function for an input of 12, and what does it represent?

Substitute 12 for r and simplify.

The value of the function for an input of 12 is 29. This means that it costs $29 to enter the carnival and take 12 rides.

C(12) = 5 + 2(12)

C(12) = 29

Example (cont.)

A carnival charges a $5 entrance fee and $2 per ride.