A College Algebra student wondered whether there could be a function such that its inverse is the same as its opposite. That is, can there be an f such that 
I had to work graphically to think this through, which you see above.
That task is now an A Assignment.
Overthinking my teaching 
So, is it cheating if I define f(x) as 0 when x = 0? Then the inverse is (0,0) and so is the opposite…
Thanks for thinking with us, Aaron. Two responses: (1) Who made me the arbiter of cheating vs. not cheating? How say you? (2) Do we need to revisit the definition of function?
On that second one, it led to a lovely exchange with a student in office hours.
Me: If a function is a set of ordered pairs such that…, then how many things do you need to have a set?
Student: Two.
I love this question! I want to try it.
You can have f(x) = ix where i is the square root of -1.
Does that work?
Oooh, mathstutor! I had not thought to consider complex functions and whether they would generate new answers to this question. Great stuff to think about there!