I'm having a bit of trouble with part of this question:

Consider real-valued functions. Is injectivity stable under addition and multiplication? What about surjectivity?

I think I understand the first part, but I start having difficulty showing surjectivity is or isn't stable under multiplication and addition. Any thoughts on how I can approach this for all real-valued functions without a particular rule on a function?

  • $\begingroup$ $f(x) = x$ and $g(x) = -x$ are both bijective, but $(f+g)(x) \equiv 0$ is neither injective nor surjective. As an exercise, it might be worth trying to come up with similar counterexamples for the multiplication case. $\endgroup$ – forgottenarrow Apr 28 at 2:52
  • $\begingroup$ Hint for a partial answer: think about multiplication by zero. $\endgroup$ – Ertxiem Apr 28 at 2:52

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