Find the function f(x) in the Functional Equation | Try this simple Trick | Math Olympiad Training

Learn how to find the function f(x) in the functional equation. Use this trick to solve this problem.

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Find the function f(x) in the Functional Equation | Try this simple Trick | Math Olympiad Training

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[fafat]

What an intelligent and awesome solution! 🤔👍🏼

[christianthomas]

hello, your answer is very smart, but I had another way to solve it as usual for functional equations.
for x = 1 we get: f(1) = -1/7 (*) ; for x=2 we get: 3f(2) + 8f(12) = 4 (**) .same for x=1/2 which gives us the equation: 3f(1/2) + 2f(2) = -7/2 (***)
Now let's suppose f is in the form f(x)=ax+b and we get f(1)=a+b= -1/7 (&) from (*), f(2) =2a+b (&&) and f(1/2) = a/2 +b .(&&&). It follows that substituting (&&) and (&&&) in (**) and (***) we get a system of two equations with two unknowns a and b, the two equations being finally: 7a + 7b = -1 and 11a + 10b = -7 after solving the system for b and using (*) you get: b=38/7 and a=-39/7; the function becoming: f(x)=(-39/7)x + 38/7

[SuperYoonHo]

Nice question. I liked the part that you got f(x) and f(1/x) because then you can just replace x by 1/x and get a new system. Then easily solve the system of equations with f(x) and f(1/x) and BINGO you got f(x)=(-39x+38)/7 !!!

[davidz]

Haven’t done many problems like this one, thank you for providing unusual (for me) math problems:)

[AA-rdnm]

Your way of solving problems are amazing. I watch your videos to brush up my 20 years old memories..Great Teacher..

[johnbrennan]

Thank you for providing us with such a wide variety of questions

[HappyFamilyOnline]

Very well explained👍
Thanks for sharing✨✨

[marioperic]

it is obvious that function is not defined in x=0. so the equation is still valid if instead of x, 1/x is taken.
thus we got:
3f(1/x)+4/x*f(x)=5/4-6.
multiplying the original equation with 3 and the second one with -4/x (because x is different than 0) we got a system:
9f(x)+12xf(1/x)=15x-18
-12xf(1/x)-16f(x)=-20+24x

summing those two equations we finally get:
-7f(x)=39x-38. so :

f(x)=-(39/7)x+38/7!

[markscher]

Since you originally substituted x by 1/x, isn’t your solution for f(1/x)?

[Hithere-wqqi]

So, the reason you replaced x with (1/×) is because it says (1/×) in the original equation? So if it was (2/×) you would replace x with (2/x) instead?

[rajeshvishwakarma]

Thanks Pre math.
I'm watching from India🇮🇳🇮🇳🇮🇳

Thanks🙏🌹❤🌹🙏

[reinamaeda]

How do we find out a specific number to multiply the whole thing with? Why is -3 picked, in this case?

[giuseppemalaguti]

f(x) =(-29/13)x+2, mi correggo f(x) =(-39/7)x+29/7....è facile fare errori di distrazione

[davidfromstow]

Fascinating - just didn't understand any of it! The fault lies with me, not the professor.

[bigm]

f(x) was a disappointed, but not you Professor Premath!

[TimBoulette]

I don't understand why you can substitute 1/x for x in equation 1. Can you explain?