Functions

• f is often times used to denote function.
• A function is an assignment of each element of set A to exactly one element of set B.
• Functions are sometimes called mappings or transformations.
• We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A.
• If at least one element in the domain has two assigned in the second set, then the function is invalid.
• Even if two domains are assigned to one in another set, the function is still valid.
• Given a function f: A -> B is read as f maps to A to B.

  Or f is a mapping from A to B.
• A -> B:

  A is called the domain of f.

  B is called the codomain of f.
• If f(a)=b,

  • b is image of a under f.
  • a is preimage of b.
• f(a)=b can also be read as b is assigned to a.
• The range of f (function) is the set of all images of points in A under f. We denote it by f(A).
• Two functions are equal when they have the same domain, same codomain, and map each element of the domain to the same element of the codomain.
• Functions can be represented in a formula.

  Example is f(x)=x+1
• A function is said to be one-to-one or injective, if and only if f(a)=f(b) implies that a=b for all a and b in the domain of f. A function is said to be an injection if each element in domain, there is a unique codomain is in its image.
• A function is said to be onto or surjective, if and only if every element in the codomain is the image of at least one element in the domain of f. A function is said to be a surjection if every element in the codomain has at least one preimage in the domain of f.
• A function is surjective if all elements in the codomain are used as image regardless if it is two or more image in the domain
• If a function is both one-to-one and onto then we say that it is bijective or a bijection.
• A function is a bijection if it is both one-to-one and onto.
• If a function is a one-to-one, then there exists an inverse.
• We can only have an inverse if f is a bijection.
• $f^-1$The inverse function maps from B to A.
• The inverse function maps our codomain to the domain.
• The floor function, denoted by $⌊x⌋$, is a mathematical function that rounds down a number to the nearest integer less than or equal to that number. In other words, it returns the largest integer less than or equal to x.
• $⌊x⌋=$$max{n∈Z∣n≤x}$ Here, Z represents the set of integers. The floor function essentially "chops off" the decimal part of a real number and leaves only the integer part.
• The ceiling function, denoted by $⌈x⌉$, is a mathematical function that rounds up a real number to the nearest integer greater than or equal to that number. In other words, it returns the smallest integer that is greater than or equal to the given number.
• For a real number x, the ceiling function is defined as: $⌈x⌉=$$min{n∈Z∣n≥x}$