For the function f(x)==2x, I get why the inverse function is f^-1(y)==y/2, but why is it ok to write it equivalently as f^-1(x)==x/2? x and y are specific objects, in this case y ==2x and x==y/2. So x and y are not equal. I do not get why you can just interchange them?
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f(x) = 2x is the same as y = 2x
To find the inverse, swap x and y and solve for y
x = 2y, so y = x/2
f(x) = 2x; f^-1(x) = x/2
What this means is that when you use a number as x, or input, in f(x), you get a value. If you then enter that value as input into f^-1(x), you get back the original x that you had entered as input into f(x). That is what the inverse of a function does. It reverses the mapping that the function did.
f(x) is a different function from f^-1(x). You could call the inverse function g(x), but that would not readily show that they are inverses. f^-1(x) is just a convenient notation to emphasise that it's the inverse of f(x).
I still do not understand. If f(x)=x and f(y)=y, then doesn't that mean that x=y?
No, it doesn't.
A function is a mapping. It takes and input and gives you an output. When we write a function, we need a placeholder for the input. We can use anything we want as the input placeholder.
If you know for a fact that \(y = 2x\), for example, then you know \(f(y) \neq f(x)\). Rather, \(f(y) = f(2x)\).