Question

Show that the functions `f(x,y)=ln x-ln y` and `g(x,y)=(x^2+2y^2)/(2xy)` are functionally dependent?

Answer 1

See below.

Explanation:

We have

`f(x,y) = ln(x/y)`
`g(x,y) = (x^2+2y^2)/(2xy)`

making `y = lambda x` we have

`f(x,y) = -ln lambda`
`g(x,y) = (1+2lambda^2)/(2lambda)`

but `lambda = e^(-f(x,y))` then

`g(x,y) = (1+2 e^(-2f(x,y)))/(2 e^(-f(x,y))) = 1/2 (e^(f(x,y))+2e^(-f(x,y)))`