Equations for Various Functions ?

[Jakes]

I don't understand the logic behind standard equation or Expressions for various Functions.

For example:
Constant Function:

so from this pic you can represent constant function is f(x) = y ..when every x value gives the same y value (7 in the above case)
constant function is written as
F(x) = y
where y is constant (it will be 7 if you go by the image image above) because every value of f(x) maps to same value in f(y)
This is Understandable.

BUT

Lets say its an Onto Function or Surjective function
if function is from X to Y then every element of Y must have some pre image in A and no element in Y should be left alone
Its equation is f(x) = y ... HOW ?

Lets take One One Function or injective function
If function is from X to Y then different elements in X have different images in Y.
Its expression is f(x) = f(y) HOW ?
[URL]http://upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg[/URL]Many One function:
(function from X to Y) If two or more different elements in X have the same image in Y
its equation or it is represented through f(x) = f(y) How ??

If the equations are wrong then please correct it

 

[Stephen Tashi]

BUT

Lets say its an Onto Function or Surjective function
if function is from X to Y then every element of Y must have some pre image in A and no element in Y should be left alone
Its equation is f(x) = y ... HOW ?

Whether a letter 'y' is a constant or a variable is always a tricky issue. When we have a function like f(x) = y = x + 2, the understanding is that 'y' is a variable. Precisely saying what "variables" are is a complicated matter.

The fact that letter denotes a variable is sometimes made clear by stating a "quantifier" such as "for each" or "there exists" that applies to it. For example, "For each ordered pair (x,y) in the function f, y = x + 2". Often no quantifers are written and the reader is supposed to understand that they are intended. For example, the statement "y = x + 2" might be written without the quantification "For each ordered pair (x,y)", but those quantifiers should be supplied by the reader.

Whether a letter is a constant is also a tricky matter. For example, a proof in geometry might begin "Let ABC be a triangle". This is understood to mean that the reader will pretend that ABC is a specific triangle, but not one that has any special properties. This happens when the writer wishes to use the principle of logic called "universal generalization" (see http://en.wikipedia.org/wiki/Generalization_(logic) for a technical treatment of this). To prove something is true "for each" triangle ABC, the writer says to consider a triangle ABC that is specific yet has no properties to distinguish it from any other triangle.

Likewise, when a writer says "c is a constant", he asks the reader to imagine that c is some specific number, but not one with special properties. (By contrast, a constant like $\pi$ is a speciific number that does have special properties.)

One statement in a book may use the symbol "y" to mean a constant (although it would be more customary to use "k" or "c" in such a role.) Another statement may use "y" to represent a variable.