Answer
$(-∞, -1] U [1, ∞) $
Work Step by Step
Given $f(x) = \sqrt[4] {x^4 - 1}$
In a even root function, the domain is defined such that the function is always at (if defined) or above 0, so $f(x) \geq 0$
$\sqrt[4] {x^4 -1} \geq 0$
$x^4 - 1 \geq 0$
$(x^2 - 1) (x^2 + 1) \geq 0$
$(x-1)(x+1)(x^2 + 1) \geq 0$
Find the zeros of the expression.
$x = 1, -1$
Test numbers in between those zero values to determine if the function is negative or positive
(-∞, -1] $(-)(-)(+) = (+)$
[-1, 1] $(-)(+)(+) = (-)$
[1, ∞) $(+)(+)(+) = (+)$
Thus the solution is $(-∞, -1] U [1, ∞) $