Introduction:
The transfer function is a mathematical representation of the relationship between the input and output of a system. It is widely used in the field of electrical engineering for analyzing and designing linear systems. In this question, we are asked to determine the applicability of the transfer function to different types of systems.

Explanation:
The transfer function is applicable to linear and time-invariant systems. Let's understand each component of this statement in detail.

Linear Systems:
A linear system is one that follows the principle of superposition and homogeneity. Superposition means that if the input to the system is a sum of multiple signals, then the output will be the sum of the individual responses to each signal. Homogeneity means that if the input is scaled by a constant factor, the output will also be scaled by the same factor. The transfer function is specifically derived and applicable to linear systems.

Time-Invariant Systems:
A time-invariant system is one whose output is not affected by a time shift in the input. In other words, if the input signal is delayed or advanced in time, the output will also be delayed or advanced by the same amount. The transfer function is derived based on the assumption of time-invariance and is applicable to such systems.

Transfer Function:
The transfer function of a system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions. It provides a concise representation of the system's behavior in the frequency domain. By analyzing the transfer function, one can determine the system's poles and zeros, frequency response, stability, and other characteristics.

Non-linear Systems:
Unlike linear systems, non-linear systems do not satisfy the principle of superposition and homogeneity. The behavior of non-linear systems is highly dependent on the specific input and can exhibit complex and unpredictable responses. Transfer functions are not applicable to non-linear systems as they cannot be represented by a simple ratio of input and output transforms.

Conclusion:
In conclusion, the transfer function is only applicable to linear and time-invariant systems. It provides a powerful tool for analyzing and designing such systems in the frequency domain. Non-linear systems, on the other hand, require different mathematical models and techniques for analysis and design.