I was reading this lecture on convex functions and I came across this
$f\colon \Bbb R^n\to \Bbb R$ is convex if and only if the function $g\colon \Bbb R\to \Bbb R$, $g(t) = f(x+tv)$, $\operatorname{dom}g=\{t\mid x+tv \mbox{ belongs to domain of }f\}$ is convex in $t$ for any $x$ belongs to the domain of $f$, $v$ belongs to $\Bbb R^n$.
So, I am a bit confused how this function of line is chosen. I mean $t$ and $v$ both are arbitrary as mentioned by the lecturer. But I want to visualize how this line looks like. I mean does it lie in the function itself.