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mathmath333
 one year ago
functions
mathmath333
 one year ago
functions

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} & \normalsize \text{if }\ f(x)\ \text{is a function satisfying} \hspace{.33em}\\~\\ & f(x)\cdot f(\frac{1}{x})=f(x)+f(\frac{1}{x}),\ \ f(4)=65 \hspace{.33em}\\~\\~\\~\\ & \normalsize \text{what will be the value of }\ \ f(6) \hspace{.33em}\\~\\ & a.)\ \ 37 \hspace{.33em}\\~\\ & b.)\ \ 217 \hspace{.33em}\\~\\ & c.)\ \ 64 \hspace{.33em}\\~\\ & d.)\ \text{none of these} \hspace{.33em}\\~\\ \end{align}}\)

alekos
 one year ago
Best ResponseYou've already chosen the best response.2anyone got an idea on how to approach this one?

alekos
 one year ago
Best ResponseYou've already chosen the best response.2i think the answer is greater than 65 maybe (b)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4\(65\) is an interesting number in the sense that \(65 = \color{red}4^3 + 1\). Is the answer \(6^3+1 \) then? I'm not exactly sure. It does hint us about the form of the function.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0looks like \(f(x)=x^n +1\) !

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Yup, that's it. And we solve for \(n\) using \(f(4) = 65\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Can we find all the functions satisfying the condition of this problem?\[f(x) f \left(\frac{1}{x} \right)=f(x)+f \left(\frac{1}{x} \right)\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4\[\left[f(x)1\right]\left[f(1/x)  1\right]=1\]

alekos
 one year ago
Best ResponseYou've already chosen the best response.2so you guys are saying that n=3 ?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4So basically any function in the form \(h(x) = f(x) + 1\) where \(f(1/x) = 1/f(x)\), right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0you mean \(h(1/x)=1/h(x)\), right?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4No, I mean that all functions \(h\) in the form \(h(x) = f(x) + 1\) satisfy this equation where \(f(1/x) = 1/f(x)\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@alekos I'm not sure yet

alekos
 one year ago
Best ResponseYou've already chosen the best response.2n=3 seems to work for f(6) = 217 and satisfies the original equation for f(1/6)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1is \(217\) the correct ans

welshfella
 one year ago
Best ResponseYou've already chosen the best response.1x^3 + 1 fits the original equation

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1in book 217 is given correct
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