functional relationships. day 1 vocabulary: a function is a relation in which each element of the...

Functional Relationships

Functional Relationships

Day 1

Vocabulary:

A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) per input (x).

x yf(x)

Sketch a linear function. Sketch a nonlinear function.

Linear Function:

Makes a line

Non-Linear Function:

Does not make a line.

How about some more definitions?

The domain is the x or input value in a function.(set of 1st coordinates of the ordered pairs)

(2, 0) or y = 3x + 2The range is the y or output value in a function.

(set of 2nd coordinates of the ordered pairs)

(2, 0) or y = 3x + 2

A relation is a set of ordered pairs.{(3, 2), (4, 2), (-2, 1)}

Given the relation {(3,2), (1,6), (-2,0)}, find the domain and range.

Domain = {3, 1, -2}

Range = {2, 6, 0}

The relation {(2,1), (-1,3), (0,4)} can be shown by

1) a table.

2) a mapping.

3) a graph.

x y2-10

134

2-10

134

How can you tell if a relation is a function without a graph? Only ONE output per input Coordinates: Check all x values. X’s can not

be repeated Mapping: Can only have one line drawn from

each x Graph: passes vertical line test

Mappingx -1 0 4 7y 3 6 -1 3

You do not need to write 3 twice in the range!

-1047

36-1

What is the domain of the relation{(2,1), (4,2), (3,3), (4,1)}

1. {2, 3, 4, 4}

2. {1, 2, 3, 1}

3. {2, 3, 4}

4. {1, 2, 3}

5. {1, 2, 3, 4}

Answer Now

What is the range of the relation{(2,1), (4,2), (3,3), (4,1)}

1. {2, 3, 4, 4}

2. {1, 2, 3, 1}

3. {2, 3, 4}

4. {1, 2, 3}

5. {1, 2, 3, 4}

Answer Now

Vertical Line Test (pencil test)

If any vertical line passes through more than one point of the graph, then that relation is not

a function.

Are these functions?

FUNCTION! FUNCTION! NOPE!

Vertical Line Test

NO WAY! FUNCTION!

FUNCTION!

NO!

Given the following table, show the relation, domain, range, and mapping.

x -1 0 4 7y 3 6 -1 3

Relation = {(-1,3), (0,6), (4,-1), (7,3)}

Domain = {-1, 0, 4, 7}

Range = {3, 6, -1, 3}

Other Related Vocabulary:

Independent Variable (input): the variable that determines the value of the

dependent variable. (x axis or domain values)

Dependent Variable (output): The variable relying on the independent variable (y

axis or range values)

EXAMPLE: the diameter of a pizza and its cost

Functional Relationships

Day 2

Finding Domain and Range of a Graph

First identify all possible values for the domain (x or input).

Next, identify all possible values for the range (y or output).

x values: -9 through +8

which can be written as: -9 ≤ x ≤ 8

y values: -3 through +8

which can be written as: -3 ≤ y ≤ 8

DOMAIN

RA

NG

E

Practice: Finding the Domain and Range of a Graph First identify all

possible values for the domain (x or input).

Next, identify all possible values for the range (y or output).

x values: -5 through +6

which can be written as: -5 ≤ x ≤ 6

y values: -4 through +7

which can be written as: -4 ≤ y ≤ 7

DOMAIN

RA

NG

E

IS THIS A FUNCTION??

Functional Relationships

Day 3

Relations & Functions-YEAR 1

A function is like a machine. You put something in and you get something out.

Sometimes equations have two variables. When there are two variables in the equation, all solutions are ordered pairs. (x, f(x))

There are an infinite number of solutions for a two variable equation.

Input

Output

Rule

f(x)

x

Function Notation

For example, with a function f(x) = 2x,

if the input is 5, then it is written as

f(5) = 2(5)

The output is ____.

Input

Output

2x

5

2(5)

10

EXAMPLE: Complete the table to find out the

human ages of dogs ages 3 through 6.

So, a 3 year old dog is 21 in human years … 4 year old dog is 28 … … 5 year old dog is 35 … … 6 year old dog is 42 …

INPUTHuman Years

RULE OUTPUTDog years

x 7x f(x)

     7(6)

7(5)

7(4)

7(3)

426

355

284

213

EXAMPLE: Make a function table to find the range of

f(x) = 3x + 5 if the domain is {-2, -1, 0, 3, 5}.

3(5) + 5

3(3) + 5

3(0) + 5

3(-1) + 5

3(-2) + 5

3x + 5

205

143

50

2-1

-1-2

f(x)x

Range: {-1, 2, 5, 14, 20}.

More Examples

EXAMPLE: Find f(-3) if f(n) = -2n – 4

EXAMPLE: Find nnff 315)( if 3

1

f(-3)= -2(-3) – 4 f(-3) = 2

3

1315

3

1f

143

1

f