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Suppose $a(x) = 2x + 5 - 3x^2 + x^3$ and $b(x) = 4 - x^2 + 4x^4$. Find $(a\circ b)(3) - (b\circ a)(3).$

 
 Jun 11, 2024
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First, let's note that \((a \cdot b) (3) = a ( b(3) ) \)

 

Now, we just work from inside to outside. We have

\(b(3) = 4 -(3)^2 + 4(3)^4 = -5 + 324 = 319 \\ a (319) = 2(319) + 5 - 3(319)^2 + (319)^3 = 32,157,119\)

 

We do the same exact thing with the second part of the equation. We get

\((b \cdot a)(3) = b (a (3)) \\ a(3) = 2(3) + 5 - 3(3)^2 + 3^3 = 11 \\ b (11) = 4 - (11)^2 + 4 (11)^4 = 58447\)

 

We have the two values needed now. All we have to do is to subtract them. We get

\((a ° b)(3) - (b ° a)(3) = 32,157,119 - 58447 = 32,098,672 \)

 

So 32,098,672 is our final answer!

 

Thanks! :)

 Jun 11, 2024

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