(x/4)^2+(2/x)^2 =(x/4)^2+2×(x/4) ×(2/x)+(2/x)^2-1 =(x/4+2/x)^2-1 Set t=x/2+4/x,t/2=x/4-2/x We get f(t)=(t/2)^2-1 Change of variable f(x)=(x/2)^2-1
@paulortega5317
Ай бұрын
let a = x/2 and b = 4/x ==> ab = 2 ==> f(a + b) = 1/b^2 + 1/a^2 = (a^2 + b^2)/(ab)^2 = [(a + b)^2 - 2ab]/4 = {(a + b)^2 - 4]/4 let u = a + b ==> f(u) = (u^2 - 4)/4
@misterdubity3073
Ай бұрын
I think that what little exposure I had to functional equations in high school involved the second method: Starting with what is inside the parentheses and trying to manipulate it to look like the right hand side. Or at least that was my intuitive way to try to solve.
@SidneiMV
Ай бұрын
x/2 = a 4/x = b ab = 2 f(a + b) = (1/a)² + (1/b)² = (1/a + 1/b)² - 2/(ab) = (a + b)²/(ab)² - 1 f(a + b) = (a + b)²/4 - 1 *f(x) = x²/4 - 1*
@Ahwke
Ай бұрын
Great❤❤❤❤❤ Great❤❤❤❤❤ Great❤❤❤❤❤
@scottleung9587
Ай бұрын
I used a method similar to your second one - my thought process was just a little different towards the end.
@achiyederi3622
Ай бұрын
What program are you using when doing the math in the videos?
@rickdesper
Ай бұрын
How I solve this: First, I look at the RHS and see that it looks related to the square of the argument for f. So, let's see what happens if I square that argument? (x/2 + 4/x)^2 = x^2/4 + 4 + 16/x^2. Looks promising, but the 4 and 16 are in the wrong places (the constant doesn't worry me, I can always subtract a constant in the last step.) But i notice if I divide everything by 4, I get the right coefficients. (1/4)(x/2 + 4/x)^2 = x^2/16 + 1 + 4/x^2 So, f(x/2 + 4/x) = (1/4)(x/2 + 4/x)^2 - 1. Or, to put in a cleaner format, f(y) = y^2/4 - 1.
@BlaqRaq
Ай бұрын
My spider sense was tingling once I saw the x’s in the numerator and the denominator but couldn’t find the idea of factoring out a 4. Hmph. Some small things trip us sometimes.
@unnegative4271
Ай бұрын
I had another way! I tried to differentiate f by chain rule! (f (x/2 + 4/x) )' = f' *(x/2 + 4/x) ' = ( x²/16 + 4/x²) ' Let y = x/2 + 4/x = (x² + 8)/2x f ' ( y ) = (x² + 8)/4x = y/2 Next integrate the f '(y) by dy f(y) = y²/4 + C y² = ( x²/4 + 4 + 16/x²) y²/4 = ( x²/16 + 1 + 4/x²) y²/4 + C = x²/16 + 4/x² so C= -1 And answer f(y) = ( y²/4 ) - 1
@phill3986
Ай бұрын
🔥👍✌️😁🔥👍✌️😁
@MathsScienceandHinduism
Ай бұрын
Do upload calculus videos too.
@vladimirkaplun5774
Ай бұрын
Not fair to promote yourself on other people channels. It might be even against the rules.
@MathsScienceandHinduism
Ай бұрын
@@vladimirkaplun5774Mind your own business
@MathsScienceandHinduism
Ай бұрын
@@vladimirkaplun5774 I know well that I can leave comments on other channels and this is not against KZitem rules. You seem to be a jobless guy with a habit of poking your nose.
@peterchan6082
Ай бұрын
I've been refraining from commenting against the use of variable t (and z) for a long time until this video. If you insist on using this letter, please use the upper case T rather than the lower case t which easily confuses with the plus sign + (especially on handwriting). Similarly z easily confuses with the number 2. Just avoid it altogether. I know you use the letters s, u, x, y, but we still have a host of other letters like m, n, p, r, v, w. Why not use them rather than z? Note that I have deliberately missed out q because it confuses with the number 9. Use Q instead.
@SyberMath
Ай бұрын
I agree
@赖皮球
Ай бұрын
f(x)=1/4(x^2-4)
@sitobubble5697
Ай бұрын
t^-(t^2-8) not equal 8 t^-(t^2-8) = x 5:51
@vladimirkaplun5774
Ай бұрын
There is a legend of Ivan Susanin (en.wikipedia.org/wiki/Ivan_Susanin) who leaded the enemies to the forest.. :) .First 8 mins
@SyberMath
Ай бұрын
?
@vladimirkaplun5774
Ай бұрын
@@SyberMath x/2 +4/x - just square it and no other ideas indeed. Your style - 8 mins of the wrong way and then quick solution
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