There is an operation on a set (real numbers) such that:

Does have an identity element with respect to operation ?

Now I find the identity element like this:

So now

Is this a valid identity element for this operation?

Because an identity element has a property that it doesn't change the original element when used with the operation on set .

can be The original element is modified.

But it's also possible that . The original element is left untouched. I'm confused by these two situations.

So my question is:

Is this a valid identity element of with respect to ?