ring of regular functions $\mathcal{O}(X)_f$ (after localization)

ring of regular functions $\mathcal{O}(X)_f$ (after localization)

Let $X$ be an affine $k$-variety, and let $f$ be an element of
$\mathcal{O}(X)$. The subvariety $D(f)$ of $X$ is a quasi-affine
$k$-variety. Is $\mathcal{O}(X)_f$ the ring of regular functions of
$D(f)$?