Question

f: R —> R, f(x) = ⌊x⌋ (Floor function) ,Examine if given function have an inverse. Find inverse, if it exists.

Answer 1

Given, $\bf{f:\mathbb{R}\rightarrow\mathbb{R},f(x)=\lfloor{x}\rfloor}$

don't confuse to see floor function , it is just greatest integer function.
Taking two different points x1 and x2 from the domain of f(x) in such a way that, f(x1) = f(x2),it will be one one only when x1 = x2

taking x1 = 4.5 and x2 = 4.3
now, f(x1) = [4.5 ] = 4
f(x2) = [4.3 ] = 4
means, f(4.5) = f(4.3) but 4.5 ≠ 4.3

hence, f is not one one function. so, f(x) is not a inversible function.