Introduction to functions

Summary

• A function with domain $A$ and codomain $B$ is a rule that for each element in $A$ assigns a unique element in B.
• We write $f: A→B$
• If $x∈A$ is an arbitrary element in the domain then $f(x)$ denotes the element in $B$ assigned by the function. This element is often denoted by $y, y=f(x)$ .
• The range of $f$ is the set of all elements in the codomain $B$ reached by the function. That is, $y$ is in the range of $f$ if $f(x)=y$ for some $x∈A$ . Formally, the range of $f$ is defined as

$$\{ y∈B \right| f\left( x \right)=y for some x∈A}$$

• The range is a subset of the codomain.
• If the codomain $B$ is $R$ or a subset of $R$ we say that the function is real-valued .
• If the domain $A$ is $R$ or a subset of $R$ we say that we have a function of a real variable .