Hahn-Banach extension of function on $\mathbb{R}^2$ with $\|\;.\,\|_1$-norm

Question

Consider $\mathbb{R}^2$ with $\|\;.\,\|_1$-norm and $M=\{(x,0) \mid x \in \mathbb{R}\}$. Define $g:M \to \mathbb{R}$ by $g(x,y)=x$. Then a Hahn-Banach extension $f$ of $g$ is given by

a) $f(x,y)=2x$

b) $f(x,y)=x+y $

c) $f(x,y)=x-2y $

d) $f(x,y)=x+2y$

Answer 1

The subspace is spanned by only one vector viz. $(1,0)$, so any linear functional on $M$ is just a real scalar. Although any linear functional on $M$ has uncountably many HB extensions.

Answer 2

See Ben Wallis's comment above:

The most obvious HB extension $f$ of $g$ is given by the rule $f(x,y)=x+y$. The other functionals all have norm $2$, and thus are not HB extensions.