Functions (Basic to Advanced)

Basics:

A function is simply a factory which takes an input and gives an output.  f(x) = 2x + 3 means, If we give 5 as input to this factory, it gives an output 2 x 5 + 3 = 13
Every function performs like this whatever type of the function we take.

What are functions? What are not?
For example in a class there are 30 students. If the teacher calls a persons roll number only one student raises his hand. Then f(roll number) is a function.
If the teacher calls a person's height then there are several students which the same height. So f(students's height) is not a function
Now take a function. y=x2$y=x^2$
For two different values of x like 2, -2 we get same value 4. So It is a function

Now what about y=x√$y=\sqrt{x}$
Do we get two values for x = 4? 2, -2
No.  x√$\sqrt{x}$ is called a principal square root fuction and takes only positive values. So it is a function.

Now what about y2=x$y^2=x$
y2=x$y^2=x$ is a combination of two functions. y=±x√$y=\pm \sqrt{x}$
So It is not a function as for x = 4, we get two values 2, -2

Check here for the graph of the function

How many of the below graphs are functions?

a. Is not a function as you draw a vertical line parrallel to y axis, it cut the curve at more than 2 points.
b. Is a function. As for each value of x, there exists only one value
c. Is not a function as for x = 0 there exists 3 values for y
d. Is not a function as discussed above
e. Is a function

Let A and B be two sets. A relation f from A to B is called a function (or a mapping or a map) from A to B if for each a∈$\in$A there exists one and only one b∈$\in$B such that the ordered pair (a, b)∈$\in$f. If (a, b)∈$\in$ f, then b is called the image of a under f. Note that two elements of A can have same image in B, but no element of A can have two images in B, and all must have some image in B. Set A is called Domain of f and set B is called co domain of f.

Total functions from A to B: 
Result: If O(A) = m and O(B) = n, then the number of possible functions from A to B is nm$n^m$ , i.e. O(B)O(A)$O(B{)}^{O(A)}$
Proof: Let A and B be two finite sets having m and n elements respectively.  m elements in A can be  mapped in n × n × .... × n (m times) = nm$n^m$ ways. Thus, the total number  of functions from A to B is nm$n^m$. 

Domain and Range:
For a function y = f(x), the set of values of x are called domain, and set of y values are called Range.