# Transfer function of an Exponential system in Z domain

Hi, I am really confused with the system in the diagram.

The input-output relation of the system is given by $y[n]=\exp(x^2[n])$

I need to find the transfer function of this system $Y(z)$ in $z$-domain. My question is: is it really the simple exponential system I am missing out or something complex. If otherwise what would be the system transfer function.

This system doesn't have a transfer function, so there's no way to find it. Only linear time-invariant (LTI) systems can be described by a transfer function. In discrete-time, the $\mathcal{Z}$-transforms of input and output sequences of an LTI system are related by
$$Y(z)=H(z)X(z)\tag{1}$$
where $H(z)$ is the system's transfer function, which is the $\mathcal{Z}$-transform of its impulse response. Equation $(1)$ is another way to state that the output sequence is given by the convolution of the input sequence with the system's impulse response. If the system is not LTI there is no function $H(z)$ which relates $X(z)$ and $Y(z)$ as in Eq. $(1)$.