The domain of the function [f(x)= ^-1(x^2-3 x+2x^2+2 x+7)] is :

Question

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

Accepted Answer

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

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

$$
\begin{aligned}
& \frac{x^{2}-3 x+2}{x^{2}+2 x+7} \leq 1 \\
& x^{2}-3 x+2 \leq x^{2}+2 x+7 \\
& 5 x \geq-5 \\
& x \geq-1
\end{aligned}
$$

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

$$
x^{2}-3 x+2 \geq-x^{2}-2 x-7
$$

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

$$
x \in R
$$

(i) $\cap$ (ii)

Domain $\in[-1, \infty)$

Solutions · 1