How do I find the domain of (x^12 - x^9 + x^4 - x + 1)^1/2 ?

Answer 1

Using the function f(x) = x¹² - x⁹ + x⁴ - x + 1, we are interested when the domain is non-negative (positive or zero)

Since even exponents will always give a non-negative solution...

x¹² + x⁴ is strictly positive for all x.

Therefore when x⁹ + x = 1 + x⁴ + x¹² is where the limit will be had...

x(x⁸ + 1) = 1 + x⁴ + x¹² --> x(x⁸ + 1) = 1 + x⁴(x⁸ + 1) --> x = 1/(x⁸ + 1) + x⁴

Since the right hand side is strictly non-negative (no odd exponents), x must be greater than zero. Define this minimal value as A. A is the root of x(x¹¹ - x⁸ + x³ -1) = -1 (with the fact the solution of x > 0)

Domain is (A,∞).