I could maybe brute-force-solve this non-integer-solution problem using a float/double, but (maybe(?)) not today. 😂

x=8 --> f(1/8)+2f(8)=3/8 x=1/8 -->f(8)+2f(1/8)=24 f(8)+2(3/8-2f(8))=24 -->f(8)=-31/4

I guess if the assumption here is that f is a function whose domain is all real numbers x except for x = 0. (Since it doesn’t say otherwise that’s a reasonable assumption) So first let’s just try x=1: f(1) + 2f(1) = 3 f(1) = 1 Now let’s try x=-1 f(-1) + 2f(-1) = -3 f(-1) = -1 So if the function was linear polynomial function (aka order 1) Then f(x) = x is one possible match so then f(8) = 8 But there could be other functions (maybe some nonlinear functions for example higher degree polynomial function functions) that satisfy the relationship/mapping for 1 and -1. So I’m stuck on how to further narrow down the possibilities of what kind of function it could be. Oh actually wait a second Let’s try 1/x then it becomes: f(x) + 2*f(1/x) = 3x And older equation for x: f(1/x) + 2*f(x) = 3/x So we treat f(x) and f(1/x) as the unknown/variables to be solved as treat everything else as constants/knowns and since it’s then a system of two equations with two unknowns maybe that will work? Multiple second equation by -2 and add to first one: -3*f(x) = 3x - 6/x f(x) = -x + 2/x Then f(8) = -31/4 Hmm that’s cool.

8

Substitute x with \dfrac1x to construct a system of linear equations in two variables, and then obtain the explicit expression for f(x).

-31/4

@grok please