Question

Let f(x) be a real valued function defined on: $R\to R$ such that $f\left(x\right)=[x{]}^2+[x+1]-3$, where [x] denotes greatest integer less than or equal to x, then which of the following option(s) is/are correct?

• A. f(x) is many-one and into function
• B. f(x) is many-one and onto function
• C. range of f(x) is $[0,\infty )$
• D. f(x) = 0 for infinite number of values of x

Answer 1

Answer: (A), (D)

The correct options are
A

f(x) is many-one and into function

D

f(x) = 0 for infinite number of values of x

$f\left(x\right)=[x{]}^2+[x+1]-3\Rightarrow (\left[x\right]+2\left)\right(\left[x\right]-1)=0$

So, $x=1,1.1,1.2,\dots \dots \Rightarrow f\left(x\right)=0$

$\therefore$ f(x) is many-one

Also, only integral values will be attained

$\therefore$ f(x) is into