First, I'll say that english isn't my first language and that I'm not studying math in english, so I'll probably say some stuff wrong.

Anyway, what I was wondering is how to find the inverse function of a function that has multiple variables? Because in $\mathbb{R}^{2}$ (in two dimensions), it's pretty simple, you can just exchange x and y and then isolate y. But is it possible in $\mathbb{R}^{3}$, in $\mathbb{R}^{4}$ or even in $\mathbb{R}^{n}$ ?

To make it clearer, take the simplest function possible, like: $z=x+y$ (another one could do the job too, I just didn't have a particular function in mind) What will be the inverse function? In my head the whole concept of inverse function in $\mathbb{R}^{n}$, when $n\gt2$, isn't clear..

If you can help me or if I'm not clear enough, let me know! Thanks!

  • $\begingroup$ You can only find the inverse of a function if it is bijective (1-1) $z=x+y$ is not bijective. i.e. $f(0,1) = f(1,0)$ While there are bijective function from $\mathbb R^2 \to \mathbb R$ they aren't as common, they are not smooth functions, and finding the inverse is not such an important activity. $\endgroup$ – Doug M Jan 31 '18 at 0:37
  • $\begingroup$ Even if you have a function from $\mathbb{R}^2 \rightarrow \mathbb{R}^2,$ finding the inverse isn't so easy. In many cases, we have to be satisfied with proving it exists. This is true even in $\mathbb{R}^1.$ We can say that $10^x$ has an inverse, but when it comes to giving a formula, all we can say is, "Let's call the inverse the logarithm." In fact, the same goes for square an square root. $\endgroup$ – saulspatz Jan 31 '18 at 0:44

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