Inverse function confusion

Question

Im reading a book published in 1914 and there is a subject: differential of an inverse funtion. example: $y=3x$ and it says $x=\frac{y}{3}$ is an inverse function. I thought if $f$ is $y=3x$ its inverse is $x=3y$ or $y=\frac{x}{3}$. Can someone explain me please. Thanks!

Answer 1

You're correct; it's ultimately a notational thing. For instance, your book solved for $x$ and left it as $$x = \frac y 3$$ Meanwhile, you did fundamentally the same thing, but swapped the roles of $x$ and $y$ after, to obtain $$y = \frac x 3$$ as the inverse function. What the book did, then, is write the inverse of the function as a function of $y$; you wrote it as a function of $x$. They're still representative of the same function, just notated differently.

Or, if $f(x) = 3x$, your book claims

$$f^{-1}(y) = \frac y 3$$

whereas you claim

$$f^{-1}(x) = \frac x 3$$

This sort of "equivalence up to your choice of how your name your variable" can be a bit confusing at first, but you'll get used to it over time.

Answer 2

Let $f$ be a function of $x$, for example, $f(x) = 3x$. What this statement means is that the value of $y$ is a function of $x$. In the example, the value of $y$ is always 3 times the value of $x$. That's why $f(x) = 3x$ and $y = 3x$ are equivalent expressions.

When we say inverse function, we don't want $y$ as function of $x$, we want $x$ as a function of $y$. This means that we want an equation which has $x$ on one side and $y$ on the other, i.e. $x = y/3$, just like your books writes. In this case, $x = y/3$ and $f^{-1}(x) = y/3$ are equivalent expressions with the exact same meaning.

Thats what an inverse function means, $x$ as a function of $y$.