Counting roots of sums of sigmoids

Question

Let $f(x)=\sum_i a_i\tanh(b_ix+c_i)+d$ be the class of sums of $n$ sigmoids parameterized by $a,b,c,$ and $d$, with all values being real. I suspect, but can't prove, that the number of roots of $f$ is at most $2n-1$. How would I go about proving this? Even the special case $n=2$ would be helpful, because it might hint at an inductive proof.