Question

If $f\left(x\right)=\sqrt{x^2+6x+9}-\sqrt{x^2-6x+9}$, then the value of $f\left(\frac{456}{123}\right)-f(-\frac{456}{123})$ is

Answer 1

$f\left(x\right)=\sqrt{(x+3{)}^2}-\sqrt{(x-3{)}^2}=|x+3|-|x-3|$
$=\{\begin{array}{cc}-6,& \text{if}x<-3\\ 2x,& \text{if}-3\le x\le 3\\ 6,& \text{if}x>3\end{array}$

$\therefore f\left(\frac{456}{123}\right)-f(-\frac{456}{123})=6-(-6)[\because \frac{456}{123}>3,-\frac{456}{123}<-3]$
$=12$