1. 4-3 Functions
A relation is a function provided there is exactly one output for each input.
It is NOT a function if one input has more than one output

2. Functions In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE OUTPUT
INPUT
Functions
(DOMAIN)
FUNCTION MACHINE
OUTPUT
(RANGE)

3. No two ordered pairs can have the same first coordinate
Example 6
Which of the following relations are functions?
R= {(9,10, (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}
No two ordered pairs can have the same first coordinate
(and different second coordinates).

4. Identify the Domain and Range. Then tell if the relation is a function.
Input Output 4
Function?
Yes: each input is mapped
onto exactly one output
Domain = {-3, 1,3,4}
Range = {3,1,-2}

5. Identify the Domain and Range. Then tell if the relation is a function.
Input Output 4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1

6. Is this a function?
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}

7. The Vertical Line Test
If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function.
Page 117

8. Use the vertical line test to visually check if the relation is a function.
(4,4)
(-3,3)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.

9. Use the vertical line test to visually check if the relation is a function.
(-3,3)
(1,1)
(3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line

10. Examples I’m going to show you a series of graphs. **don’t write 
Determine whether or not these graphs are functions.
You do not need to draw the graphs in your notes. **or write this note

11. YES!
Function? #1

12. #2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

13. #3 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

14. #4 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

15. #5
Function?
NO!

16. YES!
Function? #6
This is a piecewise function

17. Function? #7 NO! D: all reals R: [0, 1]
Another cool function: y = sin(abs(x))
Y = sin(x) * abs(x)

18. #8 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals
Another cool function: abs(x) + 2sin(x)

19. YES! #9
Function?

20. Function Notation
“f of x”
Input = x
Output = f(x) = y

21. (x, y) (x, f(x)) (input, output) y = 6 – 3x f(x) = 6 – 3x x y x f(x)
Before…
Now…
y = 6 – 3x
f(x) = 6 – 3x x y x
f(x) -2 -1 1 2 12 -2 -1 1 2 12
(x, y)
(x, f(x)) 9 9 6 6 3 3
(input, output)

22. Example.
f(x) = 2x2 – 3
Find f(0), f(-3), f(5).