Show that f : R → R, given by f(x) = x – [x], is neither one – one nor onto.

Question

Philoid · Accepted Answer

TIP: – One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

⇔ a≠b

⇒ f(a)≠f(b) for all

⇔ f(a) = f(b)

⇒ a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Now, f : A → A given by f(x) = x – [x]

To Prove: – f(x) = x – [x], is neither one – one nor onto

Check for Injectivity:

Let x be element belongs to Z i.e such that

So, from definition

⇒ f(x) = x – [x]

⇒ f(x) = 0 for

Therefore,

Range of f = [0,1] ≠ R

Hence f is not One – One function

Check for Surjectivity:

Since Range of f = [0,1] ≠ R

Hence, f is not Onto function.

Thus, it is neither One – One nor Onto function

Hence Proved