Question

The domain of the function
$f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)$
is :

• A. $[1,∞)$
• B. $[−1,2]$
• C. $[−1,∞)$
• D. $(−∞,2]$

Answer 1

Answer: (C)

$[−1,∞)$

Answer 2

$f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)$
$-1 \leq \frac{x^2 - 3x + 2}{x^2 + 2x + 7} \leq 1$
$\frac{x^2 - 3x + 2x}{2 + 2x + 7} \leq 1$
$x^2−3x+2≤x^2+2x+7$
$5x≥−5$
$x≥−1 …(i)$
$\frac{x^2 - 3x + 2}{x^2 + 2x + 7} \geq -1$
$x^2−3x+2≥−x^2−2x−7$
$2x^2−x+9≥0$
$x∈R …(ii)$
$(i)∩(ii)$
$Domain ∈ [−1,∞)$
So, the correct option is (C): $[−1,∞)$