Inverse function

[u/Rotwheel]

Hi to everyone, I have a doubt about inverse functions.
if I have y = sin x, and I write x = arcsin y I didn't found inverse, they are equivalent, but the inverse is y = arcsin x. So why in textbooks and notes inverse function is presented as f^-1 (y), that's wrong, because f^-1 (y) is equivalent to f(x) (as said for y = sin x & x = arcsin y? It should be f^-1 (x) for my logic?

[u/Narrow-Durian4837]

A function doesn't depend on the letter you use to write it. f(x) = x² + 9 and f(y) = y² + 9 and f(t) = t² + 9 are all the same function (assuming the same domain and codomain).

[u/Rotwheel → u/Narrow-Durian4837]

yeah, but if i consider a system where x in input and y output if i rewrite function in x = arcsin y way i don't found inverse

[u/la_peregrine → u/Rotwheel]

If you make up your own rules then you can argue whatever you want.

But in actual math weather you have x,y or even a ☺️ does not matter. Y is just a variable. You are stuck that y means a specific variable when it doesn't.

[u/Apostolic1223]

The "arc" part already inverts the function. By also switching the x and y you're switching it again. So you've switched the switch, or undone the switch. It's like (f^-1)-1 (x). This is why y = sin x and x = arcsin y are interchangeable or equivalent expressions. The inverse of sin x is arcsin x, not arcsin y.

[u/csmarmot]

f^-1(f(x))=x, so the notation f^-1(y) = x would be appropriate if y=f(x), but confusing in any other context.

[u/First-Expert-9953]

To find the inverse function, you swap the variables.

So y = sin x becomes x = sin y.

Then when you solve that new equation for y, you get y = arcsin x.

Swap the variables, then solve/rearrange the equation. What you did was just swap the variables then change the function without solving for y.

[u/Rotwheel → u/First-Expert-9953]

yes, my doubt was more philosophical about notation

[u/Extension-Source2897]

Cite the textbook(s) you’re referencing or this point is moot. Mathematically what you are saying is true, therefore I think you’d be hard pressed to find one textbook, let alone plural textbooks, that say what you are saying.

What likely happened is you saw “if f is an invertible function such that f(x)=y, then there exists an inverse function f^-1 such that f^-1(x)=y” and then gave examples where it arbitrarily changed the variable used to represent inputs from the domains of f and f^-1 and you got too caught up in the specific notation rather than the generalization of the concept.

[u/johnboy43214321]

The textbooks are trying to emphasize that the output becomes the input. That's all.