Final answer:
The function f(x) = x²e⁻¹x is defined for all real numbers. However, it is only positive and not zero for the interval (0, ∞).
Step-by-step explanation:
The student is asking for the interval where the function f(x) = x²e⁻¹x is defined. To find out where the function is defined, we need to look at the domain of the function, which is all the values of x that the function can take. The function f(x) is a product of two functions: x² (which is defined for all real numbers) and e⁻¹x (the exponential function, which is also defined for all real numbers). Therefore, the function f(x) is defined for all x in the real number system, (-∞, ∞). However, since we are interested in where the function is not equal to zero, we note that e⁻¹x is always positive, but x² is zero when x is zero. Hence, the interval where the function is positive (not zero) is (0, ∞).