TITLE: Notation of Elementary Inverse Functions
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: As long as you are considering $f$ and $f^{-1}$ at the same time, e.g. when discussing the continuity of $f^{-1}$ or deriving a formula for the derivative of $f^{-1}$ you should definitely keep $y$ as name for the independent variable of $f^{-1}$. But when you start discussing $f^{-1}$, say $\log$, in its own right then it might be helpful to consider it as a function of a ${\it new}\ $ horizontally scaled variable $x$.