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Let f(x) be an even function, and let g(x) be an odd function, such that f(x)+g(x)=x^2+2^x for all real numbers x. Find f(2)

 May 26, 2021
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Wow nice.

 

$f(-2)+g(-2)=4+\frac{1}{4}=\frac{17}{4}$

$f(2)+g(2)=4+4=8$

 

By definition, $f(2)=f(-2)$ & $g(-2)=-g(2)$. So sub $f(2)=f(-2)=x$ and $g(-2)=-g(2)=y$.

 

Then $x+y=\frac{17}{4}$ and $x-y=8$. Adding the two, $2x=\frac{49}{4}$ hence $x=f(2)=\frac{49}{8}$. 

 May 26, 2021

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