A function is a set of ordered pairs in which no two different ordered pairs have the same x-coordinate. An equation that produces such a set of ordered pairs defines a function.
A function is a way of dealing with an "input", applying some "rule" (the function), and then getting an "output".
We can call the input x , the rule f, and then the output is f(x), read "f of x".
This DOES NOT mean "f times x" , it's just a notation device to record the input and output.
For example, find the output of the function f(x) = x2 when the input, x = 3.
To find the output value when x = 3, substitute 3 for x in the function.
f(3) = 32
32 means 3 times 3.
(Note: f (3) is not f times 3(meaningless))
Think of f(x) = x2 as f( ) = ( )2 ; that way you can safely plug in negative numbers or even other expressions:
f(–5) = (–5)2 = 25
f(x + h) = (x + h)2
A function is a special type of relation. A relation is just a set of ordered pairs (x, y). In formal mathematical language, a function is a relation for which:
if (x1, y) and (x2, y) are both in the relation, then x1 = x2.
This just says that in a function, you can't have two ordered pairs with the same x-value but different y-values.
If you have the graph of a relation, you can use the vertical line test to find out whether the relation is a function.