Question

Find the domain of the following function:

$f\left(x\right)=\sqrt{\frac{2x+1}{x^3-3x^2+2x}}$ be $(-\infty ,k]\cup (m,n)\cup (p,\infty )$.Find $(k+m+n)\ast p$ ?

Answer 1

$f\left(x\right)=\sqrt{\frac{2x+1}{x^3-3x^2+2x}}$
$\Rightarrow \frac{2x+1}{x^3-3x^2+2x}\ge 0$
$\Rightarrow \frac{2x+1}{(x-1)(x-2)}\ge 0$
$\Rightarrow x\in (-\infty ,-\frac{1}{2}]\cup (0,1)\cup (2,\infty )$
On comparing with given domain, we get
$k=-\frac{1}{2},m=0,n=1,p=2$
$\Rightarrow (k+m+n)p=1$