Solving a Quick and Easy Functional Equation

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#ChallengingMathProblems #FunctionalEquations

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[blackpenredpen]

One of the few questions that I knew how to solve before watching your solution! And the reason I knew how to solve this type of equation is bc one time I struggled so badly when I was tutoring someone in precalc back in my college days! I would never forget about this type of equations! : )

[yvonne]

The first idea that comes to my mind is exactly this substitution because it is a common way to solve the integrals involving this denominator.
Keep up your good work!

[vkbteco]

A method that makes complicated matters easier, I also use this wonderful way.

[sppwqsr]

It must be said that f is determined only for t>0.

[math]

love how you tackle these challenging problems, appreciate your expertise

[MathElite]

Congrats on 8k subscribers!!!
Such unique questions on here, it's incredible

[luxel]

I once saw in another YouTube video that if the functional equation is of the form f(g(x))=h(x), then the solution is h(g^-1(x)), which is easy to proof. Essentially that what you did on the video, but to have it written out as an identity helps more than what one may think.

[prabu]

Put x= sinh(t). Then you will get f(e^t)= sinht /(sinht +1). On simplification you will get f(e^t) =(e^t - e^-t) /(e^t - e^-t +2). Replacing e^t by x, you get f(x).

[rotten-Z]

In 1st step t = x + √(x²+1) and x=(t² - 1)/2t
In last step t = x
How?!

[grozaiv]

Youtube algoritması sağolsun bu güzel video ile tanışmama vesile oldu. Video için teşekkürler.

[monkeyonaspacerock]

An even simpler method (assuming you're familiar with hyperbolic functions):
Let x=sinh(y)
=>
=>
=> . Let u=e^y
=> f(u) = (u^2-1)/(2u+u^2-1)
=> f(x) = (x^2-1)/(2x+x^2-1)

[joaquingutierrez]

Another way to do it is by noticing that t = e^(sinh^-1(x))

So we have x = sinh( lnt )

But what you did is better

[VSN]

Probably one of the easiest questions exploiting the nature of functions. Love your video and pls keep posting more :)

[cryfan]

Substitution all the wayyy. Amazing as always !

[carloshuertas]

Another great explanation, SyberMath!

[ernestschoenmakers]

I saw that if you plug in zero in the functional equation that this results into f(1)=0. A good check for the final function.

[fatihsinanesen]

We can solve by using trigonometric substitution. Consider a triangle with side lengths equal to 1, x and sqrt(x^2 +1). So the function becomes for the angle t. Then,

[elifotosfer]

wow, you have widened my horizon, thanks a lot!

[jofx]

When I become a math teacher, I will give this problem to my students

[ermattia]

Super video Syber, thank you! I think I learned a nice trick with this video 😁 Please, do more videos with this kind of problems!