1.6 Relations and Functions

Appetizers . . .

Can a student have more than one telephone number?

Can a student have more than one height?

Can a phone number belong to more than one person?

Some Vocabulary so we all speak the same language . . .relation: a pairing of input values with output values, such as an ordered pair. For any (x, y), x is the input and y is the output.domain: set of input values for a relation.range: set of output values for a relation.

Time of Day (PM) Temperature (°C)1 92 133 154 155 126 10

123456

913151210

Relating Time and Temperature

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7

PM

degr

ees

F

12

{(1, 9), (2, 13), (3, 15), (4, 15), (5, 12), (6, 10)}

Table

Graph

Mapping Diagram

Ordered Pairs

domain(input)

range(output)

Function or not a function?That is the question!function: there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.Huh? No repeats for x. Repeats on y are OK.

2ABC

NOT a function

ABC

2

Function

{2, A), (2, B), (2, C)} 2 is used more than once

{A, 2), (B, 2), (C, 2)} 2 is used more than once

Function or not a function?That is the question!

Function

Press Function or NOT a Function, then explain.

-2-1012

5678

XNOT a Function

Function or not a function?That is the question!

X NOT a Function

-2-1012

345

Function

Press Function or NOT a Function, then explain.

Function or not a function?That is the question!

Input Value –2 –1 0 1 2

Output Value –8 –1 0 1 8

Function XNOT a Function

Press Function or NOT a Function, then explain.

Function or not a function?That is the question!

Input Value 0 1 2 1 0

Output Value –4 –2 0 2 4

X NOT a FunctionFunction

Press Function or NOT a Function, then explain.

Function or not a function?That is the question!

X NOT a FunctionFunction

Press Function or NOT a Function, then explain.

{(1, 0), (2, –1), (3, 0), (3, – 1), (4, 0)}

Function or not a function?That is the question!

Function XNOT a Function

Press Function or NOT a Function, then explain.

{(1, 0), (2, –1), (3, 0), (4, –1)}

Vertical line test (VLT): If any vertical line passes through more than one point on the graph of a relation the relation is not a function.

What about graphs?

Function

VLT, cont.

Not a Function

VLT, cont.

Not a FunctionFunction

VLT, cont.

Not a FunctionFunction

VLT, cont.

Not a FunctionFunction

VLT, cont.

Not a FunctionFunction

VLT, cont.

Not a FunctionFunction

VLT, cont.

Not a FunctionFunction

Finding Domain and Range from a Graph

Domain = [-2, 2]

Range = [0, 2]

Helicopter taking off, hovering briefly, then landing.

Speeding baseball being caught by an outfielder.

Helicopter taking off, hovering briefly, then landing.

1.7 Function Notation

y is now f(x)

12 xy

10191331

2

2

)()()(

fxxf

functionfunction notation

Find y when x=-3; or evaluate the function when x=-3

Any letter goes!32 xy

32 xxg )(functionfunction notation

?)(?)(?)(

210

ggg