1. Relations and Functions By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: November 14, 2007

2. 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

3. Example 1 Relation  { (-4, 3), (-1, 7), (0, 3), (2, 5)} Domain  { -4, -1, 0, 2 } Range  { 3, 7, 5 } Jeff Bivin -- LZHS

4. 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

5. Example 3 Relation  y = 3x + 2 Domain  {x: x Є R } Range  {y: y Є R } Jeff Bivin -- LZHS

6. Example 4 1 5 7 0 2 11 Relation  {(1, 0), (5, 2), (7, 2), (-1, 11)} Domain  {1, 5, 7, -1} Range  {0, 2, 11} Jeff Bivin -- LZHS

7. 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

8. 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

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

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

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

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

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