Relations and Functions By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org 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 Relation  {(1, 0), (5, 2), (7, 2), (-1, 11)} Domain  5 2 7 11 -1 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? YES { (0, 2), (1, 0), (2, 6), (8, 12) } { (0, 2), (1, 0), (2, 6), (8, 12), (9, 6) } YES { (3, 2), (1, 0), (2, 6), (8, 12), (3, 5), } NO { (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) } YES { (1, 1), (1, 2), (1, 5), (1, -3), (1, -5) } NO Jeff Bivin -- LZHS

Function Operations f(x) = x2 + 2x + 1 g(x) = 3x + 2 f(x) + g(x) = Domain? f(x) + g(x) = x2 + 2x + 1 + 3x + 2 = x2 + 5x + 3 f(x) - g(x) = (x2 + 2x + 1) - (3x + 2) = x2 - x - 1 f(x) • g(x) = (x2 + 2x + 1) • (3x + 2) = 3x3 + 2x2 + 6x2 + 4x + 3x + 2 = 3x3 + 8x2 + 7x + 2 (x2 + 2x + 1) f(x) ÷ g(x) = (3x + 2) Jeff Bivin -- LZHS

Composite Functions f(x) = x2 + 2x + 1 g(x) = 3x + 2 f(g(x)) = f(3x+2) Domain? f(g(x)) = f(3x+2) = (3x+2)2 + 2(3x+2) + 1 = 9x2 + 12x + 4 + 6x + 4 + 1 = 9x2 + 18x + 9 g(f(x)) = g(x2 + 2x + 1) = 3(x2 + 2x + 1) + 2 = 3x2 + 6x + 3 + 2 = 3x2 + 6x + 5 Jeff Bivin -- LZHS

Composite Functions f(x) = x2 - 4x + 5 g(x) = x - 3 f(g(x)) = f(x-3) Domain? f(g(x)) = f(x-3) = (x-3)2 - 4(x-3) + 5 = x2 - 6x + 9 - 4x + 12 + 5 = x2 - 10x + 26 g(f(x)) = g(x2 - 4x + 5) = (x2 - 4x + 5) - 3 = x2 - 4x + 5 - 3 = x2 - 4x + 2 Jeff Bivin -- LZHS

Composite Functions f(x) = x2 + 3x + 5 f(g(x)) = f( ) Domain? f(g(x)) = f( ) = ( )2 + 3( ) + 5 = = x – 3 > 0 x > 3 Jeff Bivin -- LZHS

Composite Functions f(x) = x2 + 3x + 5 g(f(x)) = g(x2 + 3x + 5) = Domain? g(f(x)) = g(x2 + 3x + 5) = x2 + 3x + 5 = x2 + 3x + 5 - 3 = x2 + 3x + 2 > 0 (x + 2)(x + 3) > 0 -3 -2 Jeff Bivin -- LZHS