Question

f: R—>[0,1), f(x)=x-[x] ,Examine if given function have an inverse. Find inverse, if it exists.

Answer 1

f : R ------> [1, 0) , f(x) = x - [x] ,

we know, any function , f(x) is inversible in interval [a,b] only when function is one - one as well as onto.

so, Let's check function is one - one or not .
Take two different points x1 and x2 on its domain (e.g., R) such that f(x1) = f(x2) , if we get x1 = x2 after solving , then we can say that function is one - one.
f(x1) = x1 - [x1] and f(x2) = x2 - [x2]
x1 - [x1] = x2 - [x2]
x1 - x2 = [x1] - [x2]
now, you tell me if you choose any real number on R, you get, x1 = x2 , where f(x1) = f(x2)
because if we take x1 = 1.2 and x2 = -0.8
f(x1) = 1.2 - [1.2] = 1.2 - 1 = 0.2
f(x2) =- 0.8 - [-0.8]= -0.8 +1 = 0.2
here, of course f(x1) = f(x2) but x1 ≠ x2
hence, f(x) is not one one function.

hence , f(x) is not inversible function.