Hi all ---

When I tried the following question in the black oval --- the answer is in red --- I was completely stuck on the logic/thought process/intuition in the first part. I'm fine with the algebra --- it's NOT what's troubling me for the past hours.

I know how to check for symmetry about the origin and y-axis. A function, f(x), is symmetric about the origin if and about y-axis if .

But here, how on earth would you even know to check for symmetry about ? How would you even suspect or be aware of this in the first place?

For , I sort of see this because this is just the reciprocal function with the constant .My work:

So here, I'd expect symmetry about because I've seen the graph of the reciprocal function before.But I'm completely lost for?

Also, what would lead you to check for symmetry about x- and y-axis here? Question doesn't give any information that would lead you to do this?

In general, if I see this kind of question that only asks me to state all lines of symmetry (WITHOUT telling me what the lines are), how would I know

--- which lines I should check for symmetry?

--- how many lines there are?

Maybe because I don't understand it, but the answer looks like a bit of guesswork to me...

Thank you ---