How many regions are created by the graphs $y=x^3, y=x^4$ and $y=x^5$?
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Ref: R5685

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

\[y=x^3\]

\[y=x^4\]

\[y=x^5\]

$6$, (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).

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Polynomials & Rational Functions

How many regions are created by the graphs $y=x^3, y=x^4$ and $y=x^5$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

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How many regions are created by the graphs $y=x^3, y=x^4$ and $y=x^5$?
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