Is $x^{0.5}\cdot x^{0.5}$ a polynomial function on the domain for which it defines the same function as $x^1$.

Question

Is it right to say for $x>0$ it is a polynomial function? I am using the same reasoning why a constant is a polynomial e.g, $1$ defines the same function as $1x^0$ on whatever domain, so $1$ is a polynomial function defined on whatever domain you are working on.

Answer 1

Finicky details like this are probably best settled based on context more than anything. There are definite calls for working with polynomials on restricted domains, for example the Stone-Weierstrass Theorem. On the other hand, theorems such as the Fundamental Theorem of Algebra would not work if polynomials were permitted to be defined over only part of the space.

So, in short, check back on your notes, wherever polynomials are defined (as hopefully they have been). That will tell you the definition you're supposed to be working with.