The domain of the function defined by $f(x)=\frac{-5}{4 x^2+1}+\sqrt{x^2-4}$ is
The domain of the function defined by
$f(x)=\frac{-5}{4 x^2+1}+\sqrt{x^2-4}$ is
$R$
$(-\infty,-2)$
$(-\infty,-2] \cup[2, \infty)$
$(2, \infty)$
Solution
$f(x)=\frac{-5}{4 x^2+1}+\sqrt{x^2-4}$
$\because f(x)$ will be undefined if $4 x^2+1=0$ or $x^2-4 < 0$
$\because 4 x^2+1 \neq 0$ for $x \in R \quad\left\{x^2\right.$ is always positive $\}$
Now, let $x^2-4 < 0$
$\begin{array}{ll}\Rightarrow & x^2 < 4 \\ \Rightarrow & x \in(-2,2)\end{array}$
$\Rightarrow$ Function will be undefined when $x \in(-2,2)$
$\therefore$ Domain of $f(x)=R-(-2,2)$
$=(-\infty,-2] \cup[2, \infty)$