Category: Relation and Functions

Question Number 1088 by 112358 last updated on 10/Jun/15 $$Solvethefollowingintegral$$$$equationforf\left(x\right):$$$${\int }_0^{x}f\left(t\right)dt=3f\left(x\right)+k$$$$wherekisaconstant.$$ Answered by prakash jain last…

Question Number 950 by 123456 last updated on 06/May/15 $$f:\mathbb{R}\to \mathbb{R},\mathrm{continuous}\mathrm{such}\mathrm{that}\mathrm{\forall }x\in \mathbb{R},f\left(x\right)f\left[f\left(x\right)\right]=1,f\left(2004\right)=2003,f\left(1999\right)=?$$ Commented by 123456 last updated on 06/May/15 $$f\left(2004\right)f\left[f\left(2004\right)\right]=2003f\left(2003\right)=1\iff f\left(2003\right)=\frac{1}{2003}$$$$f\left(2003\right)f\left[f\left(2003\right)\right]=\frac{1}{2003}f\left(\frac{1}{2003}\right)=1\iff f\left(\frac{1}{2003}\right)=2003=f\left(2004\right)$$$${f}\left(\frac{\mathrm{1}}{\mathrm{2003}}\right){f}\left[{f}\left(\frac{\mathrm{1}}{\mathrm{2003}}\right)\right]=\mathrm{2003}{f}\left(\mathrm{2003}\right)=\mathrm{1} \…

Question Number 66462 by mathmax by abdo last updated on 15/Aug/19 $$1)simplify{S}_n\left(x\right)=\sum _{k=0}^n{C}_n^k{cos}^k\left(x\right)cos\left(kx\right)$$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{{k}}…

Question Number 863 by 123456 last updated on 30/Mar/15 $$f\left(xy\right)=f\left[xf\left(y\right)\right]$$$$f\left(x\right)=?$$ Answered by prakash jain last updated on 30/Mar/15 $$f\left(y\right)=y$$…

Question Number 821 by 123456 last updated on 17/Mar/15 $$f:\mathbb{R}\to \mathbb{R}$$$$g:\mathbb{R}\to \mathbb{R}$$$$f\left(xg\right)=f\left(x\right)g\left(y\right)$$$$g\left(xy\right)=f\left(x\right)+g\left(y\right)$$$$\frac{d\left(fg\right)}{dx}=?$$ Answered by prakash jain last…