Question

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

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

Answer 1

The correct option is C

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

$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

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

$fory\le 5\&y\ge 9y\le 5andy\ge 9Henceyϵ(-\infty ,5]\cup [9,\infty )$