Undergrad If y is a function of x, then x is also function of y?

[yosimba2000]

So then if you take the integral of y(x) dy = x dy, you get y^2 = xy.

But if x is a function of y, that means the equation is y(x)dy = x(y) dy, which gives you y^2 = X(y), where X is the integral of of x(y) dy?

 

[Lucas SV]

if $y=f(x)$, x is a function of y, i.e. $x=f^{-1}(y)$, only if $f$ is invertible. Look up inverse function theorem, for what you need.

 

[Mark44]

So then if you take the integral of y(x) dy = x dy, you get y^2 = xy.

This appears to be an intermediate step in solving a differential equation.

Starting with y dy = x dy, if you integrate the left side you get y² + a constant, but you can't integrate the right side. Is x dy a typo? Should it be x dx? If so, and we start with y dy = x dx, integrating both sides yields y² = x² + C.

But if x is a function of y, that means the equation is y(x)dy = x(y) dy, which gives you y^2 = X(y), where X is the integral of of x(y) dy?

 

[Mark44]

The short answer to your thread title -- "If y is a function of x, then x is also function of y?" -- is "Not necessarily."

For example, if y = f(x) = x², then y is a function of x, but x is not a function of y. As already stated, each variable is a function of the other only if both functions are one-to-one.