1. Function
A FUNCTION is a mathematical “rule” that for each “input” (x-value) there is one and only one “output” (y – value).
Set of Ordered Pairs:
(input, output) or (x , y)
No x-value is repeated!!!
A function has a DOMAIN (input or x-values) and a RANGE (output or y-values)
For Graphs, Vertical Line Test:
Math 3 Hon: Unit 2

2. Function Representations: f is 2 times a number plus 5
Set of Ordered Pairs: {(-4, -3), (-2, 1), (0, 5), (1, 7), (2, 9)}
Mapping:
Table x y
Graph
Function Notation:

3. Examples of a Function #1: Graphs #4: Mapping #2: Table #3: Set-2 1
- 6 8
- 4 2 x y -1 -8 3 4 7 6 13 16
{ (2,3), (4,6), (7,8), (-1,2), (0,4), (-2, 5), (-3, -2)}
Math 3 Hon: Unit 2

4. Non – Examples of a Function
#1: Graphs
#2: Table
#4: Mapping x -1 2 1 y -5 3 4 4 -2 1 8 -4 2 -1
#3: Set
{(-1,2), (1,3), (-3, -1), (1, 4), (-4, -2), (2, 0)}

5. Practice: Is it a Function?
{(2,3), (-2,4), (3,5), (-1,-1), (2, -5)}
{(1,4), (-1,3), (5, 3), (-2,4), (3, 5)} -3 4 2 1 -5 9 -2 x 3 2 5 -1 y -2 4 8
Math 3 Hon: Unit 2

6. Function Notation
Function Notation just lets us see what the “INPUT” value is for a function. (Substitution Statement)
It also names the function for us – most of the time we use f, g, or h.
Examples: f(x) = 2x
Reads as “f of x is 2 times x”
f(3) = 2 * (3) = 6
Examples: g(x) = 3x2 – 7x
Reads as “g of x is 3 times x squared minus 7 times x ”
g(-1) = 3(-1)2 – 7(-1) = 10
Math 3 Hon: Unit 2 6

7. Given f: a number multiplied by 3 minus 5
f(x) = 3x – 5
1) Find f(-4)
2) Find f(2)
3) Find f(3x)
4) Find f(x + 2)
5) Find f(x) + f(2)

8. Given g: a number squared plus 6 g(x) = x2 + 6
1) Find g(4)
2) Find g(-1)
3) Find g(2a)
4) Find 2g(a)
5) Find g(x - 1)
Math 3 Hon: Unit 2 8

9. Operations on Functions
Operations Notation:
Sum:
Difference:
Product:
Quotient:
Example 1 Add / Subtract Functions a) b)

10. Example 2 Multiply Functionsb)

11. Combining a function within another function. Notation:
Composite Function:
Combining a function within another function.
Notation:
Function “f” of Function “g” of x
“x to function g and then g(x) into function f”
Example 1 Evaluate Composites of Functions
Recall:
(a + b)2 = a2 + 2ab + b2 a) b)

12. Example 2 Composites of a Function Set
g(x)
f(x)
f(g(x)) X Y 3 5 7 9 11 X Y 7 3 5 9 8 11 4 X Y

13. Example 2 Composites of a Function Setb)
g(x)
f(x) X Y 7 3 5 9 8 11 4 X Y 5 7 3 9 11

14. Evaluate Composition Functions
Find:
f(g(3)) b) g(f(-1)) c) f(g(-4))
d) e) f)

15. Inverse Functions and Relations
Inverse Relation:
Relation (function) where you switch the domain and range values
Function  Inverse
Function  Inverse
Inverse Notation:
Inverse Properties: 1] 2]

16. Steps to Find Inverses One-to-One: [1] Replace f(x) with y
[2] Interchange x and y
[3] Solve for y and replace it with
One-to-One:
A function whose inverse is also a function (horizontal line test)
Function
Inverse
Inverse is not a function

17. Example 1 One-to-One (Horizontal Line Test)
Determine whether the functions are one-to-one. a) b)
Example 2 Inverses of Ordered Pair Relations
Are inverses f-1(x) or g-1(x) functions? a) b)

18. Inverses of Graphed Relations
FACT: The graphs of inverses are reflections about the line y = x
Find inverse of y = 3x - 2

19. Example 3 Find an Inverse Functionb) c) d)

20. Example 4: Verify two Functions are Inverses
Method 1: Directly solve for inverse and check
Method 2: Composition Property