query: Define$ f(x)$: $f(x)+f(\frac{1-x}{x} )=x$
positive_passages.docid: 142682
positive_passages.text: Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 24810
negative_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
negative_passages.docid: 127558
negative_passages.text: If $f(x)=x-\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx$ find the explicit function for $f(x)$ in its simplest form
negative_passages.docid: 98255
negative_passages.text: Find $f(x)$ such that $f(f(x)) = x^2 - 2$
negative_passages.docid: 35556
negative_passages.text: $f(z)=\frac{z}{1+|z|} $ Is 1-1?
negative_passages.docid: 106830
negative_passages.text: are $f(f(x))=f(x)$ and $f(x)=x$ equivalent?
negative_passages.docid: 67020
negative_passages.text: Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$
negative_passages.docid: 134871
negative_passages.text: Using definition of derivative to differentiate $f(x)= \sqrt{x}+1$
negative_passages.docid: 66772
negative_passages.text: $f(f(x)f(y))+f(x+y)=f(xy)$
negative_passages.docid: 112778
negative_passages.text: Let $f(x) = x^3-\frac{3}{2}x^2+x+\frac{1}{4}.$ Then the value of $ \int^{3/4}_{1/4}f(f(x))\mathrm dx$
negative_passages.docid: 44060
negative_passages.text: If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
negative_passages.docid: 122407
negative_passages.text: If $f(x) = x^3+3x+4\;,$ Then value of $ \int_{-1}^{1}f(x)dx+\int_{0}^{4}f^{-1}(x)dx$
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$
negative_passages.docid: 34790
negative_passages.text: Find $f(1)$ from equation $\int_{0}^{x}f(t)dt = x+\int_{x}^{1}tf(t)dt$
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.