Question

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

$\left|\frac{x^2+4x+2}{x^2+3}\right|\le 1$
$\iff {\left(x^2-4x+2\right)}^2\le {\left(x^2+3\right)}^2$
$\iff {\left(x^2-4x+2\right)}^2-{\left(x^2+3\right)}^2\le 0$
$\iff \left(2x^2-4x+5\right)(-4x-1)\le 0$
$\iff -4x-1\le 0\to x\ge -\frac{1}{4}$