Interesting functional equation | Can you solve ?

追加: 2026-05-21

[00:00:01]Hello, welcome back once again. Today we're going to solve the following functional equation.

[00:00:07]We're going to solve for f of x. We're given that f of x minus 4 f of negative x is equal to 1.

[00:00:16]In this video we're going to find f of x. Now, let's start by replacing x with negative x. So here, it will be f of negative x. Here, it will be f of x.

[00:00:27]Therefore, the equation will become after the substitution, that is replacing x with negative x.

[00:00:40]So the equation will become this will be f of negative x minus 4 f of x. This is equal to 1.

[00:00:52]Now, remember we're looking for f of x.

[00:00:55]So here, we have f of x here. We have f of x here.

[00:00:59]So here, we need to make sure that the coefficients of f of negative x are equal or opposite to each other.

[00:01:07]Like they may have opposite sign. Either we add or we subtract the equations and they both get eliminated. So we decide to multiply this equation by 4.

[00:01:18]So multiply this equation by 4 to get 4 f of negative x. So times the second term here with 4, negative 16 f of x.

[00:01:31]So this is equal to 1. Now, bring down the first equation.

[00:01:38]So that is f of x.

[00:01:42]Then minus 4 f of negative Oh, sorry. This here will be 4. Oh my god. So this is equal to 1.

[00:01:54]Remember we multiplied both sides here by 4, so the right answer here will be four. Now, let's see whether we're going to add or subtract the equations. Now, here, this four here is has opposite sign with this one here. So, it is best to add the equations. So, here 4 + -4 is gone. So, here, -16 f of x + f of x will give us -15 f of x. So, this is equal to 4 + 1. That will give us five. So, divide both sides by -15.

[00:02:34]So, these both get cancelled. From here, we have our solution f of x is equal to -1 over three.

[00:02:44]This right over here is our solution.

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