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

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

Relations and Functions - Part 2

Solution:

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 ]
$\therefore $ 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.