MathematicsRelations and Functions

Let f: R→R be given by f(x) = [x]^2 + [x + 1] − 3, where [x] denotes…

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

Answer & explanation

Correct answer: option 3

The correct answer is Option (2) → one-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.