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.