How to prove $f(x)=5\sqrt{x^4+1}$ is a transcedental function

Question

$$ f(x)=5\sqrt{x^4+1} $$

I know this function is transcedental function which is also not algebraic function, but i'm not sure how to prove this. Thank you for anyone to help.

Answer 1

$f$ satisfies the equation $$ f(x)^2 - 25x^4 - 25 = 0 $$ which makes it algebraic (not transcendental) by the standard definition given here. Perhaps you are thinking of a different function.

Edit: $f(x) = \sqrt[5]{x^4 + 1}$ is also algebraic. In this case, $$ f(x)^5 - x^4 - 1 = 0 $$