Question

$f\left(x\right)=\left\{\frac{|x+5|-2}{x^2+10x+21}\right\},$where {.} denotes fractional part function then which of the following is true?

• A. Number of integers in the domain of f(x) is 2.
• B. Number of integers in the domain of f(x) is 3.
• C. Range of f(x) is $(0,\frac{1}{2}]$
• D. Number of integers in the domain of f(x) is infinite.

Answer 1

The correct option is D Number of integers in the domain of f(x) is infinite.

We have,
$f\left(x\right)=\left\{\frac{|x+5|-2}{x^2+10x+21}\right\}$
Let $|x+5|=t$
$f\left(x\right)=\left\{\frac{t-2}{t^2-4}\right\}=\left\{\frac{(t-2)}{(t-2)(t+2)}\right\}$
$=\left\{\frac{1}{|x+5|+2}\right\}$
As $t\ne 2$
$\Rightarrow |x+5|\ne 2$
$\Rightarrow x+5\ne \pm 2$
$\Rightarrow x\ne -3,-7$
domain is $x\in R-\{-3,-7\}$
Range $=[0,\frac{1}{2}]-\left\{\frac{1}{4}\right\}$