Functions of several variables

Summary

• If \(f:A→B\) where \(A⊆R^n\) , \(B⊆R\) then we say that \(f\) is a real-valued function of \(n\) real variables, \(z=f(x)\) where \(x\) is the \(n\) -tuple \(x=\left( x_1,⋯,x_n \right)\) .
• If the domain is not specified, it is given by the natural domain, which is the subset of \(R^n\) for which the formula given for \(f\) is valid.
• If \(f:A→B\) is a function of \(n\) variables given by

\[f(x)=f\left( x_1,⋯,x_n \right)=a_1x_1+⋯a_nx_n+c\]

• where \(a_1,⋯,a_n\) and \(c\) are constants then \(f\) is called a linear function of \(n\) variables .