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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?

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    @tomcuchta That seems like a plausible explanation.2011-07-08

2 Answers 2

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First, the former representation is also commonly used; see, for example, Wikipedia's table here. The justification for the latter representation, however, is simply that functions are usually written in terms of $x$; see here for a concrete example.

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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 $x$.