Image of the function $y=\sqrt x$

Question

Let us take the function $$ y= \sqrt x. $$

Of course we must say that $x$ must be $\ge 0$ (domain $[0, +\infty]$). Let us suppose that we should determine its image.

I extract $x$ and obtain $$ x= y^2. $$

Now I observe that $y$ can assume any value in $\mathbb R$. BUT it is not true. $y$ is $\ge0$.

Can anyone explain me why? Where is the mistake I do?

Nick

Answer 1

The function $y=\sqrt x$ is indeed, by definition, the inverse function of the function $y=x^2$, defined in the domain $x\in [0,\infty)$ indeed assuming $x\in \mathbb R$ the latter doesn’t admit an inverse.

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