A Fun Functional Equation 

y/x=t next. y=tx next. f(x)=x[(x^2+1)^1/2]

I just mentally put one of the x's on the right under the y, put the other under the radical, saw that it was now a function only in terms of y/x and had the answer in about 5 seconds.

I solved mentally by second method

Put x=1 instead of beating around the bush

without watching: f(y/x) = y[√(x² + y²)]/x² f(y/x) = (y/x)[√(x² + y²)]/x f(y/x) = (y/x)√[1 + (y/x)²] f(x) = x√(1 + x²)

Everybody so far is wrong. The original function as written as ambiguous because it has two different values for F of -2. Depending on if y is negative or if X is negative, the result could either be positive or negative.

Set y=x^2 so f(x)=sqrt(x^2+x^4)

I went the trig route as y/x is the slope of a hypotenuse of length sqrt(x^2+y^2) Gives f(x) = x sec(arctan x) And an obscure identity later yields the same result

Could we just substitute x=1 and get the result directly?

rewrite RHS = sqrt((x^2y^2 + y^4)/x^4) -> split the addition, RHS = sqrt(y^2/x^2 + y^4 /x^4) -> let y/x=u, notice that RHS = sqrt(u^2 + u^4) = u * sqrt(1+u^2) (validate signs here) -> f(x) = x * sqrt (1+x^2)

Can you do the following? Let y=x^2 f(x) =x^2*sqrt(x^2+x^4)/x^2 =x*sqrt(1+x^2)

Take out y square out of the root. Then replace y/x with just x

We can replace y by x square

These seem dangerous and hard to know if there even is a solution without checking. If f(x+y)=x^2+y^2, then f(1+1)=2 but f(2+0)=4. Method 1 suggests that your kind of f must be constant along lines y=mx, and so the x's should cancel from the right, since there are no x's on the left.

f(y/x) =( y/x) [1+(y/x)^2]^1/2 > f(x) = x (1+x^2)^1/2.

The equation is wrong, since y*sqrt(x^2+y^2)/x^2=f(y/x)=f((-y)/(-x))=-y*sqrt(x^2+y^2)/x^2 impling that y*sqrt(x^2+y^2)/x^2=0, meaning y=0 or y=i*x

shouldn't the answer be f(x) = +-x*sqrt(x^2+1) the original equation is: f(x, y) = y*sqrt(x^2 + y^2)/x^2 "x" doesn't actually appear anywhere by itself, only "x^2" but y does appear by itself f(2/2) = f(1) = 2*sqrt(4 + 4)/4 = sqrt2 but f(-2/-2) = f(1) = -2*sqrt(4 + 4)/4 = -sqrt2

The mistake is done by taking xsqrt(x²)/x²=1 in simplifying the formula. It should be ±1 so that the answer must have a ± before it.

I used the first method, but only got one answer, which is f(x) = x*sqrt(x^2+1).

Used Second method in the first 3 seconds of this video

sounds like it could be programmed

y/x=t...f(t)=t√(1+t^2)

f(y/x)=(y/x)sqrt(1+(y/x)^2) f(x)=xsqrt(1+x^2) too easy lol

😎👍✌️😁😁✌️👍😎

It is simple. Put x=1, problem solved. 😎😎😎😎😎😎