Question

Consider a rational function $f(x)=\frac{x^2-3x-4}{x^2-3x+4}$ and a quadratic function $g(x)=x^2-(b+1)x+b-1$, where $b$ is a parameter.
The sum of integers in the range of $f(x)$, is

• A. -5
• B. -6
• C. -9
• D. -10

Answer 1

Answer: (B)

$y=\frac{x^2-3x-4}{x^2-3x+4}$
$\Rightarrow (y-1)x^2-3(y-1)x+4(y+1)=0$
Since $x\in R$, so $D\ge 0$
$\Rightarrow 9(y-1{)}^2\ge 16(y+1)(y-1)$
$\Rightarrow 7y^2+18y-25\le 0$
$\Rightarrow (y-1)(7y+25)\le 0$
$\Rightarrow \frac{-25}{7}\le y\le 1$
But $y\ne 1$, so range of $f(x)=\left[\frac{-25}{7},1\right)$
Clearly integers in the range of $f(x)$ are $-3,-2,-1,0$.
$\therefore$ Sum of integers $=-6$.