1. Relations and Functions
By: Jeffrey Bivin
Lake Zurich High School
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 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

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? 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

9. 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

10. 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 x
= 9x2 + 18x + 9
g(f(x)) =
g(x2 + 2x + 1)
= 3(x2 + 2x + 1) + 2
= 3x2 + 6x
= 3x2 + 6x + 5
Jeff Bivin -- LZHS

11. 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 x
= x2 - 10x + 26
g(f(x)) =
g(x2 - 4x + 5)
= (x2 - 4x + 5) - 3
= x2 - 4x
= x2 - 4x + 2
Jeff Bivin -- LZHS

12. 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

13. 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 =
x2 + 3x + 2 > 0
(x + 2)(x + 3) > 0 -3 -2
Jeff Bivin -- LZHS