if $$f(x)=\frac{4^{x}}{4^{x}+2}$$ , then $$f(\frac{1}{11})+f(\frac{2}{11})+f(\frac{3}{11})+\cdots+f(\frac{10}{11})=?$$ By the way, I haven't taken calculus yet.
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1What kind of an answer do you expect here? – Yuriy S Nov 22 '19 at 21:14
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The answer of the question, and any idea on how to deal with these problems. – Omar Mohamed Khallaf Nov 22 '19 at 21:16
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4Hint: $$f(1-x) = \frac{4^{1-x}}{4^{1-x}+2} = \frac{2}{4^x + 2}$$ – Brian Moehring Nov 22 '19 at 21:17
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We have that
$$f(x)=\frac{4^{x}}{4^{x}+2}=1-\frac{2}{4^{x}+2}=1-f(1-x)\implies f(x)+f(1-x)=1$$
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@John Now I understand that $f(\frac{1}{11})+f(1 - \frac{1}{11})=1$, and the answer to my question will be $5$, but how can I figure this by my own? – Omar Mohamed Khallaf Nov 22 '19 at 21:36
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1@OmarMohamedKhallaf It is a symmetry you can figure out by the graph which is true for any $f(x)=\frac{a^x}{a^x+\sqrt a}$ with $a>0$. – user Nov 22 '19 at 21:50