Answer:
A “function” is a rule which maps each element of a first set (called the “domain” of the function) to exactly one element of a second set (called the “codomain” of the function). The set of all elements in the codomain mapped-to from elements in the domain is the “range” of the function.
A function is typically defined by giving it a name, saying what it’s domain and codomain sets are, and describing the “mapping” process by means of an equation. For example, as another poster here pointed out, one can define a function by:
f:R->R such that y=f(x)=(10–5x)/7
That would constitute “y defined as a function in terms of x”. In that case, y is called “the dependent variable” and x is called “the independent variable”.
If every codomain element is mapped-to by a function from at MOST one domain element, the function is called “injective” or “one-to-one”.
If every codomain element is mapped-to by a function from at LEAST one domain element, the function is called “surjective” or “onto”. (This is equivalent to saying that the range is the entire codomain.)
A function which is both injective and surjective is said to be “bijective” or “one-to-one-and-onto”.
A function has an inverse if-and-only-if it is bijective. Also, two sets have the same “cardinality” (number of elements) if-and-only-if at least one bijection exists between them.
For example, the function f I described above is injective (because it’s strictly-increasing), and also surjective (because the y values span the entire set R), and hence it’s bijective and invertible, and its inverse is:
g:R->R such that x = g(y) = (10 - 7y)/5
Note that g(f(x))=x and f(g(y))=y. (Check it by substitution.)
Also note that g expresses “x in terms of y”, whereas f expresses “y in terms of x”.
l hope it's help
