The paper considers the boundedness character of positive solutions of the difference equation xn+1=A+xnp/xn1r, n0, where A, p, and r are positive real numbers. It is shown that (a) If p24r>4, or p1+r, r1, then this equation has positive unbounded solutions; (b) if p2<4r, or 2rp<1+r, r(0,1), then all positive solutions of the equation are bounded. Also, an analogous result is proved regarding positive solutions of the max type difference equation xn+1=max{A,xnp/xn1r}, where A, p, q(0,).