Question

Find the domain of the function $P(x) = 2024 + \sqrt{2x - 1} + 2\sqrt{x^2x} - \sqrt{x - 1}$.

Answer 1

$[1, \infty)$

Answer 2

To determine the domain, ensure all expressions under square roots are nonnegative:

1. $\sqrt{2x - 1}$ requires
   $2x - 1 \ge 0 \implies x \ge \tfrac12.$

2. $\sqrt{x^2x} = \sqrt{x^3}$ requires
   $x^3 \ge 0 \implies x \ge 0.$

3. $\sqrt{x - 1}$ requires
   $x - 1 \ge 0 \implies x \ge 1.$

The most restrictive condition is $x \ge 1$.
Therefore, the domain is

$$[1, \infty).$$