Let [ x ] denote the greatest integer [] x, where x [] R.

Question

Let [ x ] denote the greatest integer $$\le$$ x, where x $$\in$$ R. If the domain of the real valued function $$f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}} $$ is (- $$\infty$$, a) $$]\cup$$ [b, c) $$\cup$$ [4, $$\infty$$), a < b < c, then the value of a + b + c is :

Accepted Answer

For domain,

$${{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}}$$ $$\ge$$ 0

Case I :

When $${\left| {[x]} \right| - 2}$$ $$\ge$$ 0

and $${\left| {[x]} \right| - 3}$$ > 0

$$\therefore$$ x $$\in$$ (- $$\infty$$, -3) $$\cup$$ [4, $$\infty$$) ...... (1)

Case II :

When $${\left| {[x]} \right| - 2}$$ $$\le$$ 0

and $${\left| {[x]} \right| - 3}$$ < 0

$$\therefore$$ x $$\in$$ [-2, 3) ..... (2)

So, from (1) and (2) we get

Domain of function

= (- $$\infty$$, -3) $$\cup$$ [-2, 3) $$\cup$$ [4, $$\infty$$)

$$\therefore$$ (a + b + c) = -3 + (-2) + 3 = -2 (a < b < c)

$$\Rightarrow$$ Option (3) is correct.

Solutions · 1