1. FUNCTIONS

2. Function. Definition
Let X and Y be sets. A function f from X to Y is a relation from X to Y with the property that, for each element x in X, there is exactly one element y in Y such that x f y:
Since any relation from X to Y is a subset of X x Y, a function is a subset S of X x Y such that for each there is a unique with

3. Functions If f is a function from X to Y
the sets X and Y are called the domain and codomain of the function, respectively.
The unique element is called the image of under f.

4. One-to-one Function If f is a function from X to Y
and no two distinct elements of the domain are assigned the same element in the codomain, then the function is called one-to-one.
To show that a function f is one-to-one, it is necessary to show that

5. Range If f is a function from X to Y
and is a subset of the codomain such that (thus contains all the elements from Y that are paired with elements of the domain X), then is called the range of the function.

6. Onto Function
If the range and codomain of a function are equal, the function is called onto.
To show that a function f is onto, it is necessary to show that

7. One-to-one Correspondence
A function, which is both one-to-one and onto is called a one-to-one correspondence.
To show that a function f is a one-to-one correspondence, it is necessary to show that
thus, for each there is exactly one such that
means “there exists exactly one”

8. Identity Function
For any set X, the function is one-to-one correspondence. This function is called the identity function on X.

9. Composition of Functions
Let f be a function from X to Y and g be a function from Y to Z. Then it is possible to combine these two functions in a function gf from X to Z.
The function gf is called the composition of g and f and is defined by taking the image of x under gf to be g(f(x)):
If X=Z, it is also possible to define the function fg, but in general, gf≠fg

10. Inverse Function
If be one-to-one correspondence, then for each there is exactly one such that
Hence we may define a function with domain Y and codomain X by associating to each the unique such that This function is denoted by and is called the inverse of function f.

11. Inverse Function Theorem. Let is one-to-one correspondence. Then:
The inverse function of is f.
, that is

12. Exponential Function with base 2
The equation defines a function with the set of real numbers as its domain and the set of positive real numbers as its codomain. This function is called the exponential function with base 2.

13. Exponential Function with base n
In general, the equation defines a function with the set of real numbers as its domain and the set of positive real numbers as its codomain. This function is called the exponential function with base n.

14. Logarithmic Function with base n
Evidently, the exponential function with base n is a one-to-one correspondence because each element of the codomain is associated with exactly one element of the domain.
This means that the exponential function with base n has and inverse g called the logarithmic function with base n:

15. Logarithmic Function with base n
Particularly, the logarithmic function with base is the inverse of the exponential function with base 2.
The definition of an inverse function implies that if and only if :
In general, “a is true if and only if b is true” means the “a implies b” and “b implies a”.