Given the following function: $$ y = \frac{x^p}{x^p + (1 - x)^p}, $$ where $x,p\in\mathbb{R}$, I want to know which values of $p$ yields $y\in\mathbb{R}$ for all $x$. Can anyone help me with this?
Please see this link for a Desmos graph of $y$ with a slider for $p$.
Desmos seems to be using a rational approximation, $\frac ab$ where $(a,b)=1$, for the exponent, $p$. If $b$ is odd, the plot will extend to all $\mathbb{R}$. If $b$ is even, the plot will be restricted to $[0,1]$.
Desmos seems to be computing $(-1)^{a/b}=\left((-1)^{1/b}\right)^a$. When $b$ is odd, $x^b=-1$ has the real solution $x=-1$. However, when $b$ is even, $x^b=-1$ does not have a real solution.