We consider the function f from R to Z defined by fx x Is f

We consider the function f from R to Z. defined by f(x) = [x]. Is f one-to-one ? Is it onto ? We consider the function f from R to Z. defined by f (x) = [x] - [x].What is the range of f ?

Solution

f:RZ is given by, f(x) = [x]

It is seen that f(1.2)=[1.2]=1,f(1.9)=[1.9]=1.

Therefore, f(1.2)=f(1.9), but 1.21.9

Therefore, f is not one-one.

Now, consider 0.7Z.

It is known that f(x) = [x] is always an integer.

Thus, there does not exist any element xR such that f(x) = 0.7.

Therefore, f is not onto.

Hence, the greatest integer function is neither one-one nor onto