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 .