How can I tell if my cat is short sighted?

I have just taken on a rescue cat she is about 13 years old. She does not like to go outside and prefers to be in just one room in the house. She comes into the lounge to sit with me at night. She seems to wait to be approached.
She seems to respond to sound and smell more than anything else. I have to take her and put food under her nose before she realises its there. She is otherwise fit , clean and healthy. Is there anything I can do to test her sight without hurting or upsetting her.

Are you sure she isn't blind?
Try moving a silent toy in front of her and see if she reponds to it. If not, the vet would be able to tell by looking into the eye.
Best to keep her indoors if she has a sight problem as she won't be able to respond to dangers like a seeing cat.

Yes - Get some idstance away with a moving object. All cats will follow a moving object.
If the cat is sitting, chirp and make it look up. Then, move the object - like a pencil or stick back and forth, up and down and see if the cat will follow it at least with the eyes.
You can test this when the cat is in different positions. Sitting, laying or standing - and you at different elevations from directly in front to standing height above.
You can also put something at the ens of a string and let it swing. Cat should follow this. Also remember to do the up and down sets.
The cat may be going blind. It certainly is not "Short sighted" - only humans can do that. Perhaps you mean "Near sighted" and even so the can would see w moving object even if it was not sharply in focus.

13 years old for any cat is old even if she lives t for nother 6-8 years she is now officially old and her eyesight could be failing. however as she is a rescue cat what is her history? was she beaten and she is just scared? was she loved and is now upset and afraid of new surroundings?
i hope it is the latter because no matter how old the cat she will eventually trust u just give her time and her own space. remember cats go to people who hate cats as they dont tend to pester them.