Domain of f(x) when f(x 1/x) = x^2 1/x^2:

To determine the domain of the function f(x), we need to consider the values for which the function is defined. In this case, we are given that f(x 1/x) = x^2 1/x^2. Let's analyze the given expression step by step to understand its domain.

Step 1: Simplify the given expression:
The expression x 1/x can be simplified as (x*x) / x = x^2 / x = x.

Substituting this simplified expression into the given equation f(x 1/x) = x^2 1/x^2, we get f(x) = x^2 1/x^2.

Step 2: Analyze the simplified expression:
The expression f(x) = x^2 1/x^2 consists of two terms, x^2 and 1/x^2. We need to consider the domain for each term separately.

Domain of x^2:
The term x^2 represents a quadratic function, which is defined for all real values of x. Therefore, the domain of x^2 is (-∞, ∞).

Domain of 1/x^2:
The term 1/x^2 represents a rational function, which is defined for all real values of x except when the denominator is equal to zero. In this case, x^2 cannot be equal to zero, as it is always positive. Therefore, the denominator 1/x^2 is never zero, and the term is defined for all real values of x.

Step 3: Combine the domains:
Since both terms in the expression f(x) = x^2 1/x^2 are defined for all real values of x, we can conclude that the domain of the function f(x) is also (-∞, ∞).

Conclusion:
The domain of the function f(x) when f(x 1/x) = x^2 1/x^2 is (-∞, ∞). This means that the function is defined for all real values of x.