What is a vector valued function ? Vectors are just $n-tuples$ , and then how are we able to describe geometrical shapes using them ?
Is a vector function such that it outputs $n-tuples$ using other $n-$tuples and complex numbers and then you get geometrical shapes according to how you interpret these $n-tuples$ , for example ,input vectors are shifts and final output vectors and position vectors . This is only pure interpretation , if you interchanged this interpretation you won't be getting the same shape as before and in the end , it is just $n-tuples$ we are getting , and then shape is how we interpret them and nothing else ?
And if it is true , why don't books mention this ? That shapes are determined according to how we are interpreting certain $n-tuples$ in our space and we have these specific interpretations defined for this geometric space, like from now on we will interpret input vectors are shifts and final output vectors and position vectors/points and this set of points will be our plane but vectors have no role in this , it is just our interpretation in this particular case .
A vector-valued function is one where the result is a vector. In the finite-dimensional case, this is an $n$-tuple. The domain might be vectors of the same size, or of a different size, or a scalar. In the first case, one can interpret the function as a vector field. In the last case, one can interpret the function as a curve in space. All of these have geometric interpretations and are well-studied.