The domain of the real function
Question:

The domain of the real function $f(x)=\frac{x}{\sqrt{9-x^{2}}}$ is  ___________.

Solution:

Given: $f(x)=\frac{x}{\sqrt{9-x^{2}}}$

To find the domain, we find the real values of x for which the function is defined.

$x \in R$ and $9-x^{2}>0$

$\Rightarrow x \in R$ and $9>x^{2}$

$\Rightarrow x \in R$ and $x^{2}<9$

$\Rightarrow x \in R$ and $-3<x<3$

$\Rightarrow-3<x<3$

$\Rightarrow x \in(-3,3)$

Hence, the domain of the real function $f(x)=\frac{x}{\sqrt{9-x^{2}}}$ is $(-3,3)$.

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