f: R -> R
x |-> 3x + 2
let x = 5, then f maps 5 to 3*5 + 2 = 15
ie
f(5) = 15
Or we could have f(-0.27854) = 3*(-0.27854) + 2 = 1.13738
A function is also said to be one-one or injective if for each y = f(x) in the range there is only one x from the domain that maps to it. In other words if f(a) = f(b) then a function is one-one only if a=b. That is if a and b are one of the same number
To give you an example of a function that is not one-one, consider the function
f(x) = x^2
Now f(-3) = (-3)^2 =9 and f(3) = 3^2 = 9
So f(-3) = f(3)
but -3 <> 3
Hence in this case f is not a one-one function.
One way to check if a function is one-one graphically is to place a ruler horizontally through the graph. If the ruler crosses more than one point in the graph, it is not one-one.
Another term used is that of a function being onto or surjective. What this means is that if for each y in the codomain, there exists an x in the domain such that f(x) equals that y, then the function f is said to be onto. When considering the range, the function is always onto, because that is what the range is - the set of all points mapped from the domain. remember the range is a subset of the codomain.
An example of a function that is not onto is
f: R -> R
x |-> x^2
Why? because there is no number y less than 0 such that f(x) = y.
However, if we restrict the domain to the semi-open set ]-infinity, 0], then
f:R -> ]-infinity, 0]
x |-> x^2
is a surjective (onto) function
Note however, that is is still not injective (one-one).
A function that is both injective and surjective is called bijective.
Before I sign off, I'll give you an example of a function that is not defined on the whole of R.
f(x) = 1/x
f is undefined at x = 0.
So the domain of f is R-{0}, which means the set of all real numbers except 0
Furthermore there is no number x in R-{0} such that f(x) = 0. So although the codomain can be R, if it was the function would not be onto. But by restring the codomain to the range R-{0}, f becomes a surjective function. Since it is also injective, it is therefore bijective.
By the way, the function f(x) = 3x+2 is also bijective between its domain R and codomain R
Anyway, I hope that I've given you an idea of the properties and definitions of a function.