Find the inverse of f(x) = x/(sqrt(x^2+7)

This is what I did so far:

I switched the variables.

y = x/(sqrt(x^2+7))

Switch x and y,

x = y/(sqrt(y^2+7))

Square both sides,

x^2 = y^2/(y^2+7)

x^2y^2+7x^2 = y^2

Then I got the wrong answer, so I stopped here.

I was told to go on like this, which is the answer.

Though somehow it just went to 7x^2 and a 1-x^2 came along.

Like how did they do that?

y^2 = 7x^2/(1-x^2)

y = ±[7x^2/(1-x^2)]^(1/2)

Can someone explain or show in another way?

Thanks!