Category: Relation and Functions

Question Number 67519 by mathmax by abdo last updated on 28/Aug/19 $$if\frac{z}{e^z-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\frac{z^n}{n!}$$$$1)calculate{B}_0,B_1,B_2,B_3,B_4$$$$\left.\mathrm{2}\right){prove}\:{that}\:{z}\rightarrow\frac{\mathrm{1}}{{e}^{{z}}…

Question Number 1976 by 123456 last updated on 27/Oct/15 $$f:\mathbb{R}\to \mathbb{R}$$$$g:\mathbb{R}\to \mathbb{R}$$$$f(x+y)=f\left(x\right)g\left(x\right)+f\left(y\right)g\left(y\right)$$$$g(x+y)=f\left(x\right)f\left(y\right)+g\left(x\right)g\left(y\right)$$$${\left[f\left(x\right)\right]}^2+{\left[g\left(x\right)\right]}^2=?$$$$[f\left(x\right)+g\left(y\right)][g\left(x\right)+f\left(y\right)]=??$$$${f}\left({x}\right)=??? \…

Question Number 1956 by 123456 last updated on 26/Oct/15 $$x_{n+1}=y_n-1$$$$y_{n+1}=x_n+1$$$$x_0=1,y_0=1$$$${x}_{\mathrm{10}} +{y}_{\mathrm{10}} =? \…

Question Number 1873 by 123456 last updated on 19/Oct/15 $$f_{n+1}\left(x\right)=f_n\left(x\right)(x-n)(x+n)$$$$f_0\left(x\right)=1$$$$f_5\left(x\right)=?$$$$f_4\left(2x\right)=0,x=?$$ Answered by…

Question Number 132935 by liberty last updated on 17/Feb/21 Answered by EDWIN88 last updated on 17/Feb/21 $$\mathrm{let}\mathrm{x}=2\Rightarrow \mathrm{f}\left(2\right).\mathrm{f}\left(\frac{1}{2}\right)=\mathrm{f}\left(2\right)+\mathrm{f}\left(\frac{1}{2}\right)$$$$\Rightarrow 65.\mathrm{f}\left(\frac{1}{2}\right)=65+\mathrm{f}\left(\frac{1}{2}\right)$$$$\Rightarrow \mathrm{f}\left(\frac{1}{2}\right)=\frac{65}{64}\Rightarrow \mathrm{f}\left(\frac{1}{2}\right)=1+\frac{1}{64}$$$$\Rightarrow\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{6}} }\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{1}+\mathrm{x}^{\mathrm{6}} \…