Step 1: Impose the reality condition for the square root.
For f(x) = √((2 - |x|)/(3 - |x|)), the radicand must be non-negative. Hence we require (2 - |x|)/(3 - |x|) ≥ 0, along with the restriction 3 - |x| ≠ 0 to avoid division by zero.
Step 2: Introduce the substitution t = |x|.
Set t = |x| where t ≥ 0. The inequality transforms to (2 - t)/(3 - t) ≥ 0. The critical boundary points are t = 2 (numerator zero) and t = 3 (denominator zero).
Step 3: Perform sign analysis on the intervals.
For 0 ≤ t<2: both numerator and denominator are positive, yielding a positive fraction. For 2<t<3: numerator is negative while denominator remains positive, making the fraction negative. For t>3: both are negative, producing a positive fraction. At t = 2 the fraction equals zero, which is acceptable. At t = 3 the denominator vanishes, which is forbidden. The solution in t is therefore [0, 2] ∪ (3, ∞).
Step 4: Translate back to the variable x.
Since t = |x|, the condition |x| ≤ 2 implies -2 ≤ x ≤ 2. The condition |x|>3 splits into x<-3 or x>3. Combining these yields the domain (-∞, -3) ∪ [-2, 2] ∪ (3, ∞).
Step 5: Final conclusion.
The natural domain of the function is (-∞, -3) ∪ [-2, 2] ∪ (3, ∞).