Answer:
Explanation:
A relation between members of two sets X and Y is any set of ordered pairs {(x1,y1) (x2,y2) … }, finite or infinite in their number, provided the first member is from X and the second member is from Y. (Let’s not mention the uncountable possibility here).
E.g., if X = {Sunday, Monday, …, Saturday} and Y = {1,2, …, 31}, then the set {(Monday,1) (Tuesday,2) … (Monday,8) (Tuesday,9) … (Sunday,28)} is a useful relation for some February calendar, somewhere. It is “one-to-many”. And note that set Y need not be covered, but X must be. This distinguishes the domain of the relation (days of the week here) from its codomain (whole numbers 1 to 31 here).
Considering the vast set of all relations, there is one subset of great importance, those relations that have unique codomain values for each domain value. That is, each x in X is paired unambiguously with exactly one y in Y. These special relations are functions. The set of y in Y which have been paired with some x in X is called the range of f. Functions may still be “many-to-one”, as in y = r(x) which gives x rounded to the nearest integer. Thus r(2.3) = r(1.9) = 2.
Within the subset of functions are the one-to-one functions, which also have the property that every y in the range is paired with exactly one x in the domain. Because of this, these are the only functions that have an inverse function. Thus, the inverse function can be written as {(y1,x1) (y2,x2) … }. This shows that the domain and range of the original function are exactly exchanged in the inverse function.
E.g., y = tan(x) is one-to-one with domain [-pi/4 , pi/4] and range [-1,1]. Its inverse is x = arctan(y) with domain [-1,1] and range [-pi/4 , pi/4]. The roles of x and y are unambiguous, and shouldn’t be exchanged: x always in [-pi/4 , pi/4] and y always in [-1,1].