The sum, product, difference and ratio of two functions

Summary

• Given: Two real-valued functions of a real variable, \(f: A_f→B_f\) and \(g: A_g→B_g\) where \(A_f,B_f,A_g,B_g\) are subsets of \(R\)
• The sum of \(f\) and \(g\) , denoted by \(h=f+g\) , is defined as a new function , \(h:A_h→B_h\)
• The domain \(A_h\) must be a subset of \(A_f\) as well as of \(A_g\) (the natural domain is \(A_f∩A_g\) )
• The codomain must include \(B_f\) as well as of \(B_g\) (the natural codomain is \(B_f∪B_g\) )
• For \(x∈A_h\) , \(h\left( x \right)=f\left( x \right)+g(x)\)
• The product of \(f\) and \(g\) , denoted by \(h=f⋅g\) , is defined similarly with \(h\left( x \right)=f\left( x \right)⋅g(x)\)
• The difference of \(f\) and \(g\) , denoted by \(h=f-g\) , is defined similarly with \(h\left( x \right)=f\left( x \right)-g(x)\)
• The ratio of \(f\) and \(g\) , denoted by \(h=f/g\) , is defined similarly with \(h\left( x \right)=f\left( x \right)/g(x)\) . For the ratio, we must exclude real numbers that \(g\) maps to zero from the domain.