Final answer:
Determining f(x) requires understanding of even and odd functions, and the laws of exponents. The information given suggests functions involving exponentiation, with options 2) and 3) being potential candidates for f(x). However, without additional context, it is difficult to definitively choose one function as f(x).
Step-by-step explanation:
The student's question asks which function is defined as f(x). To determine this, we should look at the properties of functions and how the exponents within them behave. It's important to know that even functions multiplied with even functions result in even functions, and this same principle applies when multiplying odd functions together. However, multiplying an odd function by an even function results in an odd function.
When we have a base number (like 5 or x) raised to a power, we're engaging in exponentiation. For example, raising a number to the fourth power, n^4, means multiplying that number by itself four times. Additionally, when multiplying powers with the same base, we add the exponents (according to the laws of exponents) as shown in Eq. A.8 from the provided information.
From the options given for f(x), without additional context, it's difficult to ascertain which function precisely fits the criteria. However, if we apply the knowledge about functions and powers, we might infer that functions 2) f(x) = 3e^(2x) and 3) f(x) = 3e^(x^2) could be valid options for f(x) assuming they align with additional criteria not provided in the question. Option 1) involves a power of a power, which complicates the matter without more context, whereas option 4) is the simplest expression but doesn't incorporate the squared aspect mentioned in the supporting information.