Why is $y = \sqrt[2m+1]{x} $ defined where $ m \in \mathbb{N}, \ \forall \, x \in \mathbb{R}$ but the domain of $y = \sqrt[2m]{x} $ has to be $x≥0$ where $m \in \mathbb{N}$
To be more specific, why is negative values of $x$ allowed in former function but not in latter one?
Can you explain me the logic or what's actually going on?
$2m+1$ is always an odd number and $2m$ is always an even number (when $m\in \mathbb{N}$). You have probably heard that $\sqrt{-1}$ doesn't exist (in the set of real numbers). That's because the function $x^2$ is always non-negative. The function $x^3$, in the other hand, does take negative values, so $\sqrt[3]{-1}$ is defined. More generally, function $x^{2m}$ is always non-negative and function $x^{2m+1}$ can have any real value as output. When we reverse the functions with the help of roots, the ranges turn into domains.