Given the following function $R^2 \to R$:
$f(x,y) = e^{2x+y+1} $
Where we know that $f$ is a convex function. I am looking to define a function $g(y)$ where $g(y)$ is convex and exists in $R^2 \to R$ such that:
$g(y) = f(Ay+b) $ where A is $R^2 \to R^2$ and $b$ is a vector of dim $R^2$.
I'm unclear however how to write the function A given the function $f(x)$ as defined above.
My understanding is $g(y)$ is convex if $f(x)$ is convex using the definition of convexity.
$f(x,y) = e^{2x+y+1}$ can be written as $$ f(x,y) = \exp\left(\begin{bmatrix}2 & 1\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} + \begin{bmatrix}1 \end{bmatrix}\right) $$ interpreting the $1 \times 1$ matrix inside the exponential as a scalar. This is not exactly what you were asking for: it is the composition of an affine function $\mathbb R^2 \to \mathbb R$ (the function $2x+y+1$, which I wrote in matrix form above) and a convex function $\mathbb R \to \mathbb R$ (the function $\exp$). But that still guarantees convexity, and is the better way to set things up here.
If you really want to have an affine function $\mathbb R^2 \to \mathbb R^2$ followed by a convex function $\mathbb R^2 \to \mathbb R$, you could take the affine function $$ \begin{bmatrix}x \\y\end{bmatrix} \mapsto \begin{bmatrix}2 & 1 \\ 0 & 0\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix} + \begin{bmatrix}1 \\ 0\end{bmatrix} $$ followed up by the convex function $g(x,y) = e^x$. For that matter, the bottom row of $A$ and the bottom entry of $b$ could be absolutely anything here, though I made them $0$, because $g$ ignores its $y$-coordinate.
I don't like this as much, because checking that $g(x,y) = e^x$ is slightly more annoying than checking if a single-variable function is convex.
Of course, we can check once and for all that if a single-variable function $h : \mathbb R \to \mathbb R$ is convex, then any function $g : \mathbb R^n \to \mathbb R$ that applies $h$ to one of its coordinates, and ignores all other coordinates, is also a convex function. But the easiest way to do that is to write $g$ as a composition: the affine transformation $\mathbb R^n \to \mathbb R$ that drops the irrelevant coordinates, followed by $h$.