Let $y = f(x) = \sqrt{2x + 1}$ for $x \geq -1/2$. Then, $f$ is injective on its domain and therefore its inverse is well-defined. To find the inverse, we simply invoke the necessary algebraic operations to solve for $x$ and determine that
$ x = \frac{y^2 -1}{2} $
and therefore
$ f^{-1}(y) = \frac{y^2 -1}{2} $
Now, I realize the name of the indeterminate has no effect on the validity of the expression but in every elementary text I see, the inverse is written instead as $ f^{-1}(x) = \frac{x^2 -1}{2} $ which is really counterintuitive. If our original function maps from the "x-axis" to the "y-axis" then it makes sense that the inverse would map from the "y-axis" to the "x-axis", not conversely.
So my question is, Is there a reason why most texts choose the latter representation instead of the former or is it just a convention that is followed without any apparent justification?