How do I solve questions like these?
Okay, I'm a non maths student, so I didnt understand the solution at all. (Till now in functions Ive learned basics, composite functions, recursive, iterative functions) and non of what I learned seemed to help me here. I couldnt understand why the solution included substituting x and y.. Are there any other topics left which I need to cover in functions?
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This is not about learning a new topic, this is just how functional equations are approached. In these questions, you are not expected to “apply a formula,” you are expected to play around with the equation by substituting smart values of x and y to simplify it.
For example, people usually try values like x = 0, y = 0, x = y, or x = −y to cancel terms and extract information about f(x). That’s why the solution keeps substituting values, it’s not random, it’s done to reduce the equation step by step. So you’re not missing any concept here, you just need practice with this style of thinking. The more such questions you solve, the more natural these substitutions will start to feel.
okay...I might have a very dumb question here( forgive me, I have only studied functions to a rudimentary level) but the given relation takes two inputs f(x) and f(y)..while value has been asked of only one input f(8)...that doesnt make sense to me
Yeah I get why that feels weird at first, but that’s actually the whole point of these questions.
Even though the relation has two variables f(x) and f(y), it holds for all x and y. So you’re free to choose specific values of x and y in a smart way to simplify the equation and slowly figure out what f(x) looks like. Once you get enough information about the function, you can plug in x = 8 and get f(8).
So it’s not like a direct formula application, it’s more like use the freedom of choosing x and y to reduce the equation step by step. You’re not missing any theory here, it’s just a different style of problem that gets easier with practice.