Proving that the absolute value function is elementary.

Question

My proof is:

The absolute value function f is defined as x when $x\geq 0 $, hence it is a polynomail hence elementary function and it is defined as -x when $x < 0$ hence it is also a polynomial and hence elementary function, is my proof correct?

Answer 1

Piecewise functions are not elementary functions in general.

The absolute value can be expressed as

$$|x| = \sqrt{(x^2)}$$

which is elementary, being a composition of a polynomial root and a polynomial.

Answer 2

I don't know which concept of “elementary function” you are using, but I doubt that it allows you to deduce that if a function has several branches and it is elementary in each one of them, then it is elementary.

On the other hand, you can probably prove that the absolute function is elementary using the fact that$$(\forall x\in\mathbb{R}):\lvert x\rvert=\sqrt{x^2}.$$