On distance in total variation between image measures

We are interested in the estimation of the distance in total variation $$\Delta := \|P_{f(X)} - P_{g(X)}\|_{\mathrm var}$$ between distributions of random variables $f(X)$ and $g(X)$ in terms of proximity of $f$ and $g.$ We propose a simple general method of estimating $\Delta$. For Gaussian and trigonometrical polynomials it gives an asymptotically optimal result (when the degree tends to $\infty$).