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Question Number 107674 by Study last updated on 12/Aug/20

$$f\left(x\right)=\sqrt{x}\sqrt{x}{D}_f=???$$

Answered by Her_Majesty last updated on 12/Aug/20

$$x={re}^{i\theta };r\in {\mathbb{R}}^{+}\wedge \theta \in \mathbb{R}$$$$\Rightarrow$$$$f\left(x\right)=\sqrt{{re}^{i\theta }}\sqrt{{re}^{i\theta }}=\sqrt{r}{e}^{i\theta /2}\sqrt{r}{e}^{i\theta /2}={re}^{i\theta }=x$$

Answered by Her_Majesty last updated on 12/Aug/20

$$x⩾0\Rightarrow f\left(x\right)=x$$$$x<0\Rightarrow f\left(x\right)=i\sqrt{-x}\times i\sqrt{-x}=-1(-x)=x$$$$\Rightarrow f\left(x\right)=x$$

Commented by Study last updated on 12/Aug/20

$$whatisthedomainoff\left(x\right)?$$

Commented by Her_Majesty last updated on 12/Aug/20

$$ifwestayin\mathbb{R}thedomainisx⩾0$$

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