Question

Find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if $x$ is real

• A. $(-\infty ,5]\cup [9,\infty )$
• B. $(-\infty ,-5]\cup [9,\infty )$
• C. $[–5,9]$
• D. $[5,9]$

Answer 1

The correct option is A

$(-\infty ,5]\cup [9,\infty )$

$y=\frac{x^2+34x-71}{x^2+2x-7}$

$\Rightarrow x^2(y-1)+x(2y-34)+(71-7y)=0$

Given that $x$ is real.

$\Rightarrow b^2-4ac\ge 0\Rightarrow (2y-34{)}^2-4(y-1\left)\right(71-7y)\ge 04[y^2+289-34y+7y^2-78y+71]\ge 0y^2-14y+45\ge 0y(y-9)-5(y-9)\ge 0(y-5\left)\right(y-9)\ge 0$

$y\le 5$ & $y\ge 9$

Hence $y\in (-\infty ,5]\cup [9,\infty )$.