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