>>11901529
You have a function, say f. Similarly to how it depended on a single variable if the domain was [math]\mathbb{R}[/math], it can now depend on 2 variables, say x and y. At no point do you need to define a direct relation between x and y, and in fact when you start seeing partial derivatives you'll see that the other variables are usually treated as constants, because as far as x is concerned y is just another number there, they are not involved with one another (unless you define them to be that way).
You can define a basic operation like the sum using this. For example, [math]f(x,y) = x+y[/math]. This is just the usual sum, and as you may know picking an arbitrary summand doesn't lock your choice for the second summand, they are not dependent of one another.
What you defined before indeed just sounds like you are mixing it up with function composition. Also, now it seems like you are actually talking about vector-valued functions, where the multiple components don't come in the variables but in the images. If you had a vector valued function of one component, you could actually write it like [math]f(x) = (a(x),b(x),c(x))[/math], which I think is what you meant. In that case, there is only one variable and each component function is still independent of one another. The functions a,b,c here are the usual real valued functions you are probably used to, and they can be defined however you want. You can even mix both notions and create a vector valued multivariable function, and in that case each component of the image would be a multivariable function that is defined by (for example) two variables, and in fact that's how you will usually do stuff like parametrizing a surface in [math]\mathbb{R}^{3}[/math], for example.