Relations and Functions

By: Sri Elniati

Last Updated: November 14, 2007

Definitions

Relation A set of ordered pairs.

Domain The set of all inputs (x-values) of a relation.

Range The set of all outputs (y-values) of a relation.

Jeff Bivin -- LZHS

Example 1

Relation { (-4, 3), (-1, 7), (0, 3), (2, 5)}

Domain { -4, -1, 0, 2 }

Range { 3, 7, 5 }

Jeff Bivin -- LZHS

Example 2

Relation

{ (-2, 2), (5, 17), (3, 3), (5, 1), (1, 1), (7, 2) }

Domain { -2, 5, 3, 1, 7 }

Range { 2, 17, 3, 1 }

Jeff Bivin -- LZHS

Example 3

Relation y = 3x + 2

Domain {x: x Є R }

Range {y: y Є R }

Jeff Bivin -- LZHS

Example 4

1

5

7

-1

0

2

11

Relation

{(1, 0), (5, 2), (7, 2), (-1, 11)}

Domain {1, 5, 7, -1}Range {0, 2, 11}

Jeff Bivin -- LZHS

Definition

Function A relation in which each element of the domain ( x value) is paired with exactly one element of the range (y value).

Jeff Bivin -- LZHS

Are these functions?

{ (0, 2), (1, 0), (2, 6), (8, 12) }

{ (0, 2), (1, 0), (2, 6), (8, 12), (9, 6) }

{ (3, 2), (1, 0), (2, 6), (8, 12), (3, 5), }

{ (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) }

{ (1, 1), (1, 2), (1, 5), (1, -3), (1, -5) }

Jeff Bivin -- LZHS

Function Operationsg(x) = 3x + 2f(x) = x2 + 2x + 1

f(x) + g(x) =

f(x) - g(x) =

f(x) • g(x) =

x2 + 2x + 1 + 3x + 2

(x2 + 2x + 1) - (3x + 2)

= x2 + 5x + 3

= x2 - x - 1

(x2 + 2x + 1) • (3x + 2)= 3x3 + 2x2 + 6x2 + 4x + 3x + 2= 3x3 + 8x2 + 7x + 2

f(x) ÷ g(x) = (x2 + 2x + 1)

(3x + 2)

Domain?

32,: xRxx

Rxx :

Rxx :

Rxx :

Jeff Bivin -- LZHS

Composite Functionsg(x) = 3x + 2f(x) = x2 + 2x + 1

f(g(x)) =

g(f(x)) =

= (3x+2)2 + 2(3x+2) + 1

= 3(x2 + 2x + 1) + 2

= 9x2 + 12x + 4 + 6x + 4 + 1

= 3x2 + 6x + 3 + 2

Domain?

Rxx :

Rxx :

f(3x+2)

= 9x2 + 18x + 9

g(x2 + 2x + 1)

= 3x2 + 6x + 5Jeff Bivin -- LZHS

Composite Functionsg(x) = x - 3f(x) = x2 - 4x + 5

f(g(x)) =

g(f(x)) =

= (x-3)2 - 4(x-3) + 5

= (x2 - 4x + 5) - 3

= x2 - 6x + 9 - 4x + 12 + 5

= x2 - 4x + 5 - 3

Domain?

Rxx :

Rxx :

f(x-3)

= x2 - 10x + 26

g(x2 - 4x + 5)

= x2 - 4x + 2Jeff Bivin -- LZHS

Composite Functionsf(x) = x2 + 3x + 5

f(g(x)) = = ( )2 + 3( ) + 5

=

Domain?

,3

f( )

=

Jeff Bivin -- LZHS

3)( xxg

3x3x 3x

5333 xx332 xx

x – 3 > 0

x > 3

Composite Functionsf(x) = x2 + 3x + 5

g(f(x)) =

=

= x2 + 3x + 5 - 3

Domain? g(x2 + 3x + 5)

=

Jeff Bivin -- LZHS

3)( xxg

3

232 xx

,23, x2 + 3x + 2 > 0

(x + 2)(x + 3) > 0

-2-3

x2 + 3x + 5