Question

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

Answer 1

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

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

$$\begin{gathered} x\in R\mathrm{and}9-x^2>0 \\ \Rightarrow x\in R\mathrm{and}9>x^2 \\ \Rightarrow x\in R\mathrm{and}{x}^2<9 \\ \Rightarrow x\in R\mathrm{and}-3<x<3 \\ \Rightarrow -3<x<3 \\ \Rightarrow x\in \left(-3,3\right) \end{gathered}$$

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