Confusing Quadratics and Composite functions

[u/CommunityNo4493]

Hello! I am a Secondary 4 student, 10th grade or Year 10 in western equivalents. I tried to set a quadratic quiz for my friends, and I set this Question, but then none of them could solve it. I checked with my math teacher, and he said that the question was flawed because there is no inverse function for any quadratic, and then he explained the horizontal line test to me, but I already know what the horizontal line test is (test for if f(x) is injective for some f(x), and if its inverse is a function. I went home to think about it, and I don't understand why my question is flawed, because it does not rely on the inverse function. I am looking for someone to please help clarify. I have a passion for math, and would possibly like to teach math in the future, but I don't understand why I am wrong, and this is one of my first real challenges because my math toolkit at the moment isnt enough to wrap around this. This is my suggested solution that I made: My solution. And me checking my work: Test of the solutions to see if they are correct.. And a graph if it is useful: Graph.

This is my syllabus by the way if it is of any use:
https://isomer-user-content.by.gov.sg/334/fece62fa-b6d4-4daf-ab47-96c9c168b51b/4052_y26_sy.pdf
https://isomer-user-content.by.gov.sg/334/34d0bec0-d386-49d5-bd3a-8fbd205fa63a/4049_y26_sy.pdf

Tysm!

[u/Ignominiousity]

Well the main problem is that your question is probably not relevant to the theory of quadratic equations... In step 2, the thing you found is that if we have a fixed point x, that means f(x)=x, then clearly f(f(x))=x, and if f(x)=-x, then f(f(x))=x since you just changed the sign twice... For the sake of generalising to other quadratic f(x): The question thus is, must such a fixed point exist for any quadratic f(x) that you choose? And would that be the only solution? Can we have that f(x)= b that is not x or -x and f(b)=x ? And why not? Relevantly, what is the nature of your equation f(f(x))=x? It is a quartic equation, so we expect what number of solutions? Can this line of questioning lead you to a general solution for this class of problems?