The function f: R → R defined by f(x) = [x], ∀ x ∈ R , is

Question

The function f: R → R defined by f(x) = [x], ∀ x ∈ R , is (A) one-one (B) onto (C) Both one-one and onto (D) neither one-one nor onto

Tardigrade Admin · Accepted Answer

Let f(x1) = f(x2) ⇒ [x1] = [x2] ⇒ x1 = x2 [For example, if x1 = 1.4 and x2 = 1.5, then [1.4] = [1.5] = 1 ] ∴ f is not one-one . Also, f is not onto as its range (set of integers) is a proper subset of its co-domain R.

Q. The function $f : R \to R$ defined by $f (x) = [x], \forall x \in R$ , is

one-one

onto

Both one-one and onto

neither one-one nor onto

Let $f (x_{1}) = f (x_{2}) \Rightarrow [x_{1}] = [x_{2}] \Rightarrow x_{1} = x_{2}$
[For example, if $x_{1} = 1.4$ and $x_{2} = 1.5$ , then [1.4] = [1.5] = 1 ]
$∴$ f is not one-one .
Also, f is not onto as its range (set of integers) is a proper subset of its co-domain R.

Q. The function f:R→R defined by f(x)=[x],∀x∈R , is