Functions
A function is a relation between numbers which assigns exactly 1 y or f(x) to each x.
x does not repeat with a different y.
y can repeat or not repeat without changing anything.

x did NOT repeat. We don’t care that y repeated, even if it repeats and repeats.
This IS a function 

x did repeat. This is NOT a function. 
Look at the charts, Decide if the following are functions or NOT..
x 
f(x) 

x 
f(x) 
 2 
 1 
 2 
 7 

0 
0 
1 
2 

 2 
1 
0 
2 

 8 
2 
 1 
2 

 18 
 3 
3 
5 
The chart above left is NOT a function. x repeated.
The chart above right IS a function. x did not repeat.
x DOES NOT repeat, it is a FUNCTION;
x repeats—NOT a function.
SIMPLE if you don’t get mixed up.
Can be shown as sets of ordered pairs.
1^{st } in parenthesis is x, 2^{nd} in parenthesis is y,
{(2, 3), (3, 4), (4, 5)} 
x does NOT repeat with a different y IS a FUNCTION 
{(2, 3), (1, 4), (2, 5)} 
x repeats with different y’s. This is NOT a function 
Decide if the set represents a function or NOT: { (2, 1), (5, 1), (2, 2)} 

{ (0, 4), (1, 5), (5, 6)} 
The first above IS NOT a function.
The second IS a function.
Relations can be shown as mappings.
Look at the mappings,
if x does NOT go to more than 1 y, it is a function.
More than one x can go to one y,
BUT one x CANNOT go to more than 1 y.
If only 1 arrow originates from each x , it is a function
If relations are shown as graphs,
Imagine a vertical line
is passed from far left to far right of each graph.
If that line touches more than one point at any time,
the graph is not a function.
Domain is what x is. Range is what y is.
y can be written as f(x), read f of x
What is the range of y = 2x^{2} – 5 if the domain is { – 2, 0, 1}? Substitute the domain in for x, simplify to find y, y is the range.
y = 2x^{2}  5 y =2(– 2)^{2}  5 y = 2(4) – 5 y = 8 – 5 y = 3 
y = 2x^{2}  5 y =2(–0)^{2}  5 y = 2(0) – 5 y = 0 – 5 y = – 5 
y = 2x^{2}  5 y =2(1)^{2}  5 y = 2(1) – 5 y = 2 – 5 y = – 3 
range is { – 5, – 3, 3) if put into numerical order
What is the range of this function?
{( – 3, 4), (0, 0), (1, – 2), ( 3, 2)}
range is what y is, the 2^{nd} number in each ordered pair.
4, 0,  2 , 2
SO {– 2, 0, 2, 4} in order. SIMPLE
If the relation is shown on a graph ,
range is what y ranges from (lowest point) and to (highest point) no matter where they occur in the graph
It is beyond my ability to show the graphs and mappings on this web, but I have them on an office document.