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 $y=\frac{x^2+2x+1}{x^2+2x-1}$
$⟹y{x}^2+2xy-y=x^2+2x+1$
$⟹(y-1)x^2+2x(y-1)-(y+1)=0$
$B^2-4AC\ge 0\&y\ne 1,y\ne 0$