Composite Functions

Question

Is the composite function g ◦ f is a constant function if: f(x) = [x] and g(x) = x − [x], where [x] denotes the largest integer less than or equal to x ?

Answer 1

Yes the given composite function is a constant function. 

f(x) = [x] GIF

g(x)= x-[x] = {x} Fractional part function 

GoF = {[x]} 

We know [x] always give integer, and we also that fractional part of integer is 0.

Hence gof always give 0 as answer. 

Answer 2

f(x) = [x], g(x)  = x-[x]

(gof) (x) = g(f(x)) = f[x] = [x] - [[x]]

 = [x] - [x]

 = 0

Hence, gof is a zero function which is a constant function.