Are all polynomials rational functions? Are all algebraic functions polynomials?

It is known that all polynomials are not rational functions.
Example:
Consider the function $f(x)=\pi$. This, function is a polynomial but it cannot be written in the form of a rational function.
Hence, all polynomials are not rational functions.
Step 2
No, not all algebraic functions are polynomials.
If an algebraic expression has a radical, then it cannot be a polynomial.