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What do you mean by the term max function

Big-oh has some useful properties. Some of them are listed as theorems here. Let use start with the definition of max function.

Definition (max function): Let f_{1}(x) and f_{2}(x) be functions from a set A to a set of real numbers B. Then max( f_{1}(x) , f_{2}(x) ) is the function from A to B that takes as its value at each point x the larger of f1(x) and f2(x).

Theorem 2: If f_{1}(x) is O( g_{1}(x) ) , and f_{2}(x) is O( g_{2}(x) ) , then (f_{1} + f_{2})( x ) is O( max( g_{1}(x), g_{2}(x) ) ) .

From this theorem it follows that if f_{1}(x) and f_{2}(x) are O( g(x) ) , then (f_{1} + f_{2})( x ) is O( g(x) ), and

(f_{1} + f_{2})( x ) is O( max( f_{1}(x) , f_{2}(x) ) ) .

Theorem 3: If f_{1}(x) is O( g_{1}(x) ) , and f_{2}(x) is O( g_{2}(x) ) , then (f_{1} * f_{2})( x ) is O( g_{1}(x) * g_{2}(x) ) .