Question

f: R—>Z, f(x) =⌈x⌉ (Ceiling function) ,Examine if given function have an inverse. Find inverse, if it exists.

Answer 1

Ceiling function is the least integer which is greater than or equal to x. it is denoted by ceil(x) or $\lceil{x}\rceil$
for example :- $\lceil{2.4}\rceil=3$

f : R ----->Z , f(x) = $\lceil{x}\rceil$
f(x) will be inversible only when f(x) is one one as well as onto function.

Let's check function is one one or not.
take two points x1 = 1.2 , x2 = 1.4 in its domain {e.g., R} such that f(x1) = f(x2) , if we get , x1 = x2 in f(x1) = f(x2) we can say f(x) is one one function otherwise it is many one.

f(x1) = f(1.2) = $\lceil{1.2}\rceil=2$
f(x2) = f(1.4) = $\lceil{1.4}\rceil=2$
here , f(x1) = f(x2) but x1 ≠ x2
hence, it is clear that function is not one one.

hence, function is not inversible function.