The concept of a function is that of something which gives a unique output for a given input.

Definition 1.2.1 Consider two sets, D and R along with a rule which assigns a unique element of R to every element of D. This rule is called a function and it is denoted by a letter such as f. Given x ∈ D, f

is the name of the thing in R which results from doing
f to x. Then D is called the domain of f. In order to specify that D pertains to f, the
notation D

may be used. The set R is sometimes called the range of f. These days it
is referred to as the codomain. The set of all elements of R which are of the form f

for some x ∈ D is therefore, a subset of R. This is sometimes referred to as the image of
f. When this set equals R, the function f is said to be onto, also surjective. If whenever
x≠y it follows f

≠f

, the function is called one to one. , also injective It is common
notation to write f : D

R to denote the situation just described in this definition where
f is a function defined on a domain D which has values in a codomain R. Sometimes you
may also see something like D

R to denote the same thing.

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