Question

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos ^{-1}\left(\frac{x^2-4 x+2}{x^2+3}\right)$ is :

• A. $\left(-\infty, \frac{1}{4}\right]$
• B. $\left[-\frac{1}{4}, \infty\right)$
• C. $\left(-\frac{1}{3}, \infty\right)$
• D. $\left(-\infty, \frac{1}{3}\right]$

Answer 1

Answer: (B)

$ \left|\frac{x^2+4 x+2}{x^2+3}\right| \leq 1$
$ \Leftrightarrow \left(x^2-4 x+2\right)^2 \leq\left(x^2+3\right)^2$
$\Leftrightarrow \left(x^2-4 x+2\right)^2-\left(x^2+3\right)^2 \leq 0 $
$\Leftrightarrow \left(2 x^2-4 x+5\right)(-4 x-1) \leq 0 $
$ \Leftrightarrow -4 x-1 \leq 0 \rightarrow x \geq-\frac{1}{4}$