Objectives:1. To determine if a

relation is a function2. To find the domain

and range of a function

3. To evaluate functions

• As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook.

Relation Function

Input Output

Domain Range

Set-Builder Notation Interval Notation

Function Notation

A mathematical relationrelation is the pairing up (mapping) of inputs and outputs.

A mathematical relationrelation is the pairing up (mapping) of inputs and outputs.

• DomainDomain: the set of all input values• RangeRange: the set of all output values

A toaster is an example of a functionfunction. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

“What comes out of a toaster?”“It depends on what you put in.”– You can’t input bread and expect a waffle!

A functionfunction is a relation in which each input has exactly one output.

• A function is a dependent relation

• Output depends on the input

RelationsRelations

FunctionsFunctions

A functionfunction is a relation in which each input has exactly one output.

• Each output does not necessarily have only one input

RelationsRelations

FunctionsFunctions

If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIGBIG trouble.

Darth Vadar as a “Procurer.”

Tell whether or not each table represents a function. Give the domain and range of each relationship.

The size of a set is called its cardinalitycardinality. What must be true about the cardinalities of the domain and range of any function?

Which sets of ordered pairs represent functions?1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

A relation is a function iff no vertical line intersects the graph of the relation at more than one point

FunctionFunction Not a FunctionNot a Function

If it does, then an input has more than one output.

To determine if an equation represents a function, try solving the thing for y.

• Make sure that there is only one value of y for every value of x.

Determine whether each equation represents y as a function of x.

1. x2 +2y = 4

2.(x + 3)2 + (y – 5)2 = 36

Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder set-builder notationnotation.

Examples:• {x | x < -2} reads “the set of all x such that x is

less than negative 2”.

Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder set-builder notationnotation.

Examples:• {x : x < -2} reads “the set of all x such that x is

less than negative 2”.

Another way to describe an infinite set of numbers is with interval notationinterval notation.

• ParenthesisParenthesis indicate that first or last number is notnot in the set:– Example: (-, -2) means the same thing as x < -2– Neither the negative infinity or the negative 2 are

included in the interval– Always write the smaller number, bigger number

Another way to describe an infinite set of numbers is with interval notationinterval notation.

• BracketsBrackets indicate that first or last number is in the set:– Example: (-, -2] means the same thing as x -2– Infinity (positive or negative) never gets a bracket– Always write the smaller number, bigger number

• DomainDomain: All x-values (L → R)– {x: -∞ < x <

∞}

• RangeRange: All y-values (D ↑ U)– {y: y ≥ -4}

Domain: All real numbers

Range: Greater than or equal to -4

Determine the domain and range of each function.

• DomainDomain: What you are allowed to plug in for x.– Easier to ask what you can’t plug in for x.– Limited by division by zero or negative even roots– Can be explicit or implied

• RangeRange: What you can get out for y using the domain.– Easier to ask what you can’t get for y.

Determine the domain of each function.1. y = x2 + 2

2. 2

19

yx

Determine the domain of each function.1.

2.

2y x

2 2y x

Functions can also be thought of as dependent relationships. In a function, the value of the output dependsdepends on the value of the input.

• Independent quantityIndependent quantity: Input values, x-values, domain

• Dependent quantityDependent quantity: Output value, which depends on the input value, y-values, range

The number of pretzels, p, that can be packaged in a box with a volume of V cubic units is given by the equation p = 45V + 10. In this relationship, which is the dependent variable?

In an equation, the dependent variable is usually represented as f (x).

• Read “f of x”– f = name of function; x = independent variable– Takes place of y: y = f (x)– f (x) does NOT mean multiplication!– f (3) means “the function evaluated at 3” where

you plug 3 in for x.

Evaluate each function when x = -3.1. f (x) = -2x3 + 5

2. g (x) = 12 – 8x

Let g(x) = -x2 + 4x + 1. Find each function value.• g(2)

• g(t)

• g(t + 2)

Objectives:1. To determine if a

relation is a function2. To find the domain

and range of a function

3. To evaluate functions

Assignment:Continue

Pgs 118-119#47-79 odd