The function f(x) = (x - 2)/(x + 1) , x ≠ -1 , is

The correct answer is Option (1) → strictly increasing in \((-\infty, -1)\) and strictly increasing in \((-1, \infty)\)

By the quotient rule:

    \( \displaystyle f'(x) = \frac{(x + 1) - (x - 2)}{(x + 1)^{2}} \)

    \( \displaystyle = \frac{3}{(x + 1)^{2}} \)

The square \((x + 1)^{2}\) is positive for every x ≠ −1, so \(f'(x) \gt 0\) on both sides of x = −1.

So f is strictly increasing in \((-\infty, -1)\) and also in \((-1, \infty)\).