For a vector valued function $\mathbf{f}: X \subset \mathbb{R}^n \to \mathbb{R}^m$, the definition of differentiability suggests that if $\mathbf{f}$ is differentiable at point $\mathbf{a} \in X \subset \mathbb{R}^n $, then for $ \mathbf{x}$ near $\mathbf{a}$, $\mathbf{f}(\mathbf{x})$ can be approximated by the tangent hyperplane in $\mathbb{R}^m$: $$ \mathbf{f}(\mathbf{a}) + D_{\mathbf{f}}(\mathbf{a})(\mathbf{x}-\mathbf{a}) = \mathbf{f}(\mathbf{a}) + \sum_i^n \frac{\partial}{\partial x_i}\mathbf{f}(\mathbf{a})(x_i-a_i) $$
This is the parametric equation of a hyperplane in $\mathbb{R}^m$, the co-domain.
A special case of $\mathbf{f}$ is the scalar valued function $f: \mathbb{R}^n \to \mathbb{R}$, with the equation of the tangent hyperplane: $$ f(\mathbf{a}) + D_{f}(\mathbf{a})(\mathbf{x}-\mathbf{a}) = f(\mathbf{a}) + \nabla f(\mathbf{a})(\mathbf{x}-\mathbf{a}) $$ We see this as the point-normal form of a hyperplane in $\mathbb{R}^{n+1}$ (domain and co-domain combined) , and it is tangent to the graph of $f$, a subset of $\mathbb{R}^{n+1}$ defined by $\{(x_1,...,x_n,x_{n+1}) | x_{n+1}=f(x_1,...,x_n)\}$. This is quite different from the situation above of $\mathbf{f}$ (how do you define the graph of $\mathbf{f}$?), and I'm not sure what the special case of $m=1$ would mean (what does a hyperplane in $\mathbb{R}^1$ look like?).
How do I reconcile these two interpretations?