Suppose I have a function $f(x)$ that is defined on $\{0\}\cup[1/2,1]$ such that $f(0)=0$ and $f(x)=1+x$ for $1/2\leq x\leq1$.
I want to define the following extension of this function that is defined on $[0,1]$: $\hat{f}(x)=f(x)$ on $\{0\}\cup[1/2,1]$ and $\hat{f}(x)=2x$ on $(0,1/2)$ so that for each of the points in between $0$ and $1/2$, the graph of the new function is just the convex hull of $(0,f(0))$ and $(1/2,f(1/2))$.
Is there a name for this type of operation?
It's sometimes called piecewise linear extension. A longer form is extending $f$ in a piecewise linear way, which may be more suitable when the original function has nonlinear pieces.
(Some say piecewise affine instead of piecewise linear, to emphasize it's $y=ax+b$ and not just $y=ax$. But I think in the presence of "piecewise" confusion is unlikely.)