Question

For real values of $x$, the range of $\frac{x^2+2x+1}{x^2+2x-1}$ is

• A. $(-\infty ,0)\cup (1,\infty )$
• B. $[\frac{1}{2},2]$
• C. $(-\infty ,\frac{-2}{9}]\cup (1,\infty )$
• D. $(-\infty ,-6]\cup (-2,\infty )$

Answer 1

The correct option is A $(-\infty ,0)\cup (1,\infty )$

Let $y=\frac{x^2+2x+1}{x^2+2x-1}$

$\Rightarrow y{x}^2+2xy-y=x^2+2x+1$

$\Rightarrow (y-1)x^2+2x(y-1)-(y+1)=0$

$B^2-4ac\ge 0$ & $y\ne 1,y\ne 0$