Question

Consider a rational function $f(x)=\frac{x^2-3 x-4}{x^2-3 x+4}$ 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^2-3 x-4}{x^2-3 x+4} $
$ \Rightarrow(y-1) x^2-3(y-1) x+4(y+1)=0$
Since $x \in R$, so $D \geq 0$
$\Rightarrow 9( y -1)^2 \geq 16( y +1)( y -1) $
$\Rightarrow 7 y ^2+18 y -25 \leq 0$
$\Rightarrow( y -1)(7 y +25) \leq 0 $
$\Rightarrow \frac{-25}{7} \leq y \leq 1$
But $y \neq 1$, 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 $=-6$.