We construct an infinite family of maps on F₂[[q]] that are conjugate to the famous 3x + 1 map on the 2-adic integers. We identify one such map that has the further property that the positive integers whose 3x + 1-orbit contains 1 correspond to polynomials via a conjugacy, and whose polynomial orbits all eventually enter the 2-cycle or one of the two fixed points. Proving that every positive integer corresponds to a polynomial via this conjugacy would settle the original 3x + 1 conjecture.