Into how many regions is the plane divided when the following equations are graphed, not considering the axes?




  1. \(6\), (b) \(7\), (c) \(8\), (d) \(9\), (e) \(10\).

A careful sketch seems like our best plan, noting that all three curves go through \((0,0)\) and \((1,1)\), and that \(x^3>x^4>x^5\) on the interval \(0<x<1\).

We also have that \(y=x^3\) and \(y=x^5\) both go through \((-1,-1)\), and \(x^3 < x^5 < 0\) on the interval \(-1<x<0\).

So here is our sketch, without including the axes;

a sketch of the three curves y equals x cubed, y equals x to the power four, and y equals x to the power five

We have nine regions, and so the answer is (d).