1.4 Relations & Functions

Relation: a set of ordered pairs

Domain (D): set of first coordinates of the pairs

Range (R): set of second coordinates of the pairs

Correspondence between domain & range is mapping

Special mapping is a function!

(usually x-value)

(usually y-value)

Function: f ; a relation in which each element in the domain is mapped to exactly one element in the range

can we put this in our own words??...

Relations & functions can be expressed in different ways

one popular one: f (x)

such as f (x) = 2x2 + 7

*remember to evaluate f (x) for a particular value of x,

simply replace the value for x every time you see an x

or g(x) = cos x

Ex 1) Given , evaluate: a)

b)

c)

22( 2) 2( ) 3( )2m

2( ) 2 3m x x x 2(4) ( 6) 8 6 14

2( ) 2( ) 3( )m h x hx x h 2 22( 2 ) 3( )x xh h x h 2 22 4 2 3 3x xh h x h

2 2( ) ( ) 2( ) 3( ) (2 3 )x h xm x h m x x xh

h h

2 2 22 4 2 3 3 2 3x xh h x h x x

h

24 2 3xh h h

h

(4 2 3)4 2 3

h x hx h

h

*this one is something

special…but we don’t want to spill the secret quite yet!

If a function has no domain specified, it is “All Real #s” (R)You have to find domain?

Ex 2) Find Domain & Range (confirm with graph)

a)

b)

Remember – No dividing by 0 – No taking even roots of neg. #s

2

1( )g x

x

so…x ≠ 0

2 0x 2{ : 0}D x x

{ : 0}R y y

2( )

5

xh x

x

5 0x

so…x < 5{ : 5}D x x { : }R y y R

Graph on Calculator

Graph on Calculator

Two functions are equal if

(1) the domain of f is equal to domain of g AND

(2)for all x in domain, f (x) = g(x)

Ex 3) Are these functions equal?

NO…. Domains are same but… h(x) ≠ m(x)

22

1( ) 3 and ( )

3h x x m x

x

2

3

x

Determining if a relation / rule / mapping is a function…Vertical Line Test!!

all vertical lines drawn only touch graph 1 time – then YES!

Ex 4) Are these functions?a)

one-to-one function?passes horizontal line test

y = 5 1y

x and y x y x

b) c)

(basically, no y’s repeat)

YES! YES! NO!NO! N/A (not a function)

Homework#104 Pg 29 #1-23 odd, 24-27, 28, 35, 37,

38, 43, 44, 45, 49, 53, 55, 57