Concept: Smallest integer function:(Ceiling function) The function f (x) = is called the smallest integer function and the value of will always be lesser than or equal to the inner function. ⇒ ≥ x. If f :A → B and g : C → D. Then (fog) (x) will exist if and only if the co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C. Here, the codomain of f is Z and the domain of g is also Z so (gof) (x) is defined. The graph of the smallest integer function, , where denotes smallest integer function and g(x) = 5x Here, we have to find the value of g o f(1/2) ⇒ g o f(1/2) = g( f(1/2)) ∵ f(x) = so, f(1/2) = = = 0 ⇒ g o f(1/2) = g(0) ∵ g(x) = 5x so, g(0) = 50 = 1 Hence, g o f(1/2) = 1