According this website eigne function is defined as

http://mirlab.org/jang/books/audioSignalProcessing/dssEigenFunction.asp?title=9-4%20Eigen%20Functions%20(%A9T%A6%B3%A8%E7%BC%C6)

If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system.

but in this question its saying only Option c is eigen function.

my Question is why cant it be Option b and d as it will also obey the defintion of eigne function as given in the website?

PS: Please let me if know my question is not clear i will edit it but please don't downvote.

According to this website:

If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system.

but in this question it is saying only option $(c)$ is eigen function.

My question is why can't it be option $(b)$ and $(d)$ as it will also obey the defintion of eigen function as given in the website?

PS: Please let me if know my question is not clear i will edit it but please don't downvote.