Let $f: R→R$ be given by $f(x) = [x]^2 + [x + 1] - 3$, where [x] denotes the greatest integer less than or equal to x. Then, f(x), is

many-one and into

We have,

$f(x) = [x]^2 + [x + 1] - 3$

$⇒f(x) = [x]^2 +[x]+1-3$   [$∵[x+n] = [x]+n$, where $n ∈ Z$]

$⇒f(x)=[x]^2+[x]-2$

$⇒f(x) = ([x] + 2) ([x] −1)$

Clearly, $f(x) = 0$ for all $x ∈ [1, 2)∪[-2, -1)$. So, f is a many-one function.

Also, f(x) assumes only integral values.

∴ Range of $f ≠ R$.

Hence, f(x) is a many-one into function.