Why am I wrong about this Gradient? Help

Question

$f(x,y) = \langle 3x+y+1,2x-y-1 \rangle = A\vec{x}-\vec{b}$

Why are we unable to take the gradient of $f(x,y)$?

Is this a trick question. I can take the partial derivative of each function

$\frac{\partial f_x\:}{\partial \:x}\left(\left(3x+y\right)\right)=3$

$\frac{\partial f_y\:}{\partial \:y}\left(\left(2x-y\right)\right)=-1$

So why is the gradient not?

$=\begin{bmatrix}3\\ -1\end{bmatrix}$

Answer 1

The gradient of a real-valued function $f: \mathbb{R^2} \to \mathbb{R}$ is defined as $\nabla f = \langle f_x,f_y\rangle$. Your function is not a scalar-valued function hence gradient is not applied here.