Why is the graph of the function $y=x^{2.9}$ not tangible on the negative side

Question

Why is the graph of $y=x^{2.9}$ intangible on the negative side?

Answer 1

In $\mathbb{R}$, one can define $x^r$ with $r\notin\mathbb{Z}$ in two ways.

1. Define, for all real number $r$ that is not an integer, $$x^r=e^{r\ln{x}}$$ where $e$ is the exponential base and $\ln$ the natural logarithm.

As the natural logarithm is not defined when $x\leq0$ this means that the function $x\mapsto x^r$ with $r\notin\mathbb{Z}$ is only defined on $\mathbb{R}^+_\star$ if $r<0$, or $\mathbb{R}^+$ if $r>0$ (because of limit of $\exp(t)$ when $t\rightarrow-\infty$).

2. However there is a way to accept the case $x<0$ : if $r\in\mathbb{Q}$ with $r=\dfrac{1}{q}$ and $q$ odd, one can define $x^{1/q}$ as the unique real number $y$ (and possibly negative) where $y^q=x$. This can be extended to $r=\dfrac{p}{q}$ if $p\wedge q=1$ and $q$ is odd (this is how Geogebra works, actually).

If $r=2.9=\dfrac{29}{10}$, the denominator is even so $x$ can't be negative