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  •  Considering only the principal values of the inverse trigonometric functions, the domain of the function <img alt="\mathrm{f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{2}+3}\right)}" src="

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

Option: 1

\mathrm{\left(-\infty, \frac{1}{4}\right]}


Option: 2

\mathrm{\left[-\frac{1}{4}, \infty\right)}


Option: 3

\mathrm{(-1 / 3, \infty)}


Option: 4

\mathrm{\left(-\infty, \frac{1}{3}\right]}


Answers (1)

\begin{aligned} &\mathrm{\left|\frac{x^{2}+4 x+2}{x^{2}+3}\right| \leq 1} \\ &\mathrm{\left(x^{2}-4 x+2\right)^{2} \leq\left(x^{2}+3\right)^{2}} \\ & \mathrm{\left(x^{2}-4 x+2\right)^{2}-\left(x^{2}+3\right)^{2} \leq 0} \\ & \mathrm{\left(2 x^{2}-4 x+5\right)(-4 x-1) \leq 0} \\ & \mathrm{-4 x-1 \leq 0 \rightarrow x \geq-\frac{1}{4}} \\ & \mathrm{{\left[-\frac{1}{4}, \infty\right)}} \end{aligned}

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Ramraj Saini

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