First slide

Theory of expressions

Question

Let $f (x) = \frac{x^{2} - 2 x + 3}{x^{2} - 2 x - 8}, x \in R - {- 2, 4}$
The range of $f$ is

Moderate

A

$(\frac{- 2}{9}, 1]$
B

$R - (\frac{- 2}{9}, 1)$
C

$(- \infty, \frac{- 2}{9}]$
D

$R - (\frac{- 2}{9}, 1]$

Solution

Let $y = \frac{x^{2} - 2 x + 3}{x^{2} - 2 x - 8}, x \in R - {- 2, 4}$
Note that y π 1.
Now,
$(y - 1) x^{2} - 2 x (y - 1) - (8 y + 3) = 0$
As $x$ is real $∆ \geq 0$
$\begin{array}{l} 4 (y - 1)^{2} + 4 (y - 1) (8 y + 3) \geq 0 \\ \Rightarrow (y - 1) [y - 1 + 8 y + 3] \geq 0 . \\ \Rightarrow (y - 1) (y + 2 / 9) \geq 0 \\ \Rightarrow y \leq - 2 / 9 or y > 1 . [∵ y \neq 1] \\ Thus, y \in R - (\frac{- 2}{9}, 1] \end{array}$

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