Given z=f(x,y), how would I go about finding the inverse function(range) like it was done for f(x)=y,when f(x)=x^2-1 ,for example,we would put x^2-1=y and solve for x?

Given z=f(x,y), how would I go about finding the inverse function(range) like it was done for f(x)=y,when f(x)=x^2-1 ,for example,we would put x^2-1=y and solve for x?

 

I don't know what you mean by "inverse function (range)" since "inverse function" and "range" are very different things. In general, functions of two variables do not have inverse functions since they are seldom "one to one". Finding the range of a function of two variables depends strongly on exactly what the function is.

Inverse and range of a function are different things.

Inverse functions existe in more than one variable and are in general far more complicated or impossible to find.

The range of a function in more than two variables are functions of the other variables.

Example: f(x,y)=log(x+y), for x=8 we find y=(-8,+inf). The range function is x+y>0, so x>-y and y>-x.