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

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Inverse Trigonometric Functions

The domain of the function

f (x) = sin^{- 1} (\frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7})

is :

Answer: (C)

For

sin^{- 1}

to be defined, we need

- 1 \leq \frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7} \leq 1

. The denominator

x^{2} + 2 x + 7 = (x + 1)^{2} + 6 > 0

for all real

x

. For the upper bound:

\frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7} \leq 1

gives

x^{2} - 3 x + 2 \leq x^{2} + 2 x + 7

, so

- 5 x \leq 5

, i.e.,

x \geq - 1

. For the lower bound:

\frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7} \geq - 1

gives

x^{2} - 3 x + 2 \geq - (x^{2} + 2 x + 7)

, so

2 x^{2} - x + 9 \geq 0

, which holds for all real

x

(discriminant

< 0

). Thus the domain is

x \in [- 1, \infty)

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Single correct option