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.

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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 offunction?On that second one, it led to a lovely exchange with a student in office hours.

Me: If a function isa 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!