Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 145879 by mathdanisur last updated on 09/Jul/21

Answered by mr W last updated on 09/Jul/21

$$letu=f\left(f\left(x\right)\right)$$$$f\left(u\right)⩽0$$$$\Rightarrow \frac{1-u}{1+u^2}⩽0$$$$\Rightarrow 1-u⩽0$$$$\Rightarrow u⩾1$$$$letv=f\left(x\right)$$$$u=f\left(v\right)⩾1$$$$\Rightarrow \frac{1-v}{1+v^2}⩾1$$$$\Rightarrow 1-v⩾1+v^2$$$$\Rightarrow v^2+v⩽0$$$$\Rightarrow v(v+1)⩽0$$$$\Rightarrow -1⩽v⩽0$$$$\Rightarrow -1⩽f\left(x\right)⩽0$$$$\Rightarrow -1⩽\frac{1-x}{1+x^2}⩽0$$$$\Rightarrow -1-x^2⩽1-x⩽0$$$$\Rightarrow x^2-x+2⩾0\wedge x⩾1$$$$x^2-x+2⩾0isalwaystrue.$$$$\Rightarrow x⩾1\Leftarrow solution$$

Commented by iloveisrael last updated on 09/Jul/21

$$\mathrm{v}(\mathrm{v}+1)⩾0\Rightarrow \mathrm{v}⩽-1\cup \mathrm{v}⩾0$$$$$$

Commented by iloveisrael last updated on 09/Jul/21

$$\mathrm{v}⩽-1\cup \mathrm{v}⩾0$$$$\mathrm{f}\left(\mathrm{x}\right)⩽-1\cup \mathrm{f}\left(\mathrm{x}\right)⩾0$$$$\frac{1-\mathrm{x}}{1+{\mathrm{x}}^2}⩽-1\cup \frac{1-\mathrm{x}}{1+{\mathrm{x}}^2}⩾0$$$$\Rightarrow 1-\mathrm{x}⩽-{\mathrm{x}}^2-1\cup 1-\mathrm{x}⩾0$$$$\Rightarrow {\mathrm{x}}^2\underset{\mathrm{x}=\varnothing }{\underbrace{-\mathrm{x}+2⩽0}}\cup \mathrm{x}⩽1$$$$$$

Commented by mr W last updated on 09/Jul/21

$$noneedtobesofastsir!$$$$yougivecommentsosoonandi{don}^{\prime }t$$$$evenhavesomesecondstofixtypos...$$$$trytounderstandatfirstwhatidid.$$$$iletu=f\left(f\left(x\right)\right),then$$$$f(f\left(f\left(x\right)\right)⩾0meansf\left(u\right)⩾0,i.e.$$$$\frac{1-u}{1+u^2}⩾0$$$$.....$$

Commented by mr W last updated on 09/Jul/21

Commented by mr W last updated on 09/Jul/21

$$now{it}^{\prime }scompletewithmyworking.$$$$nowyoucanbegintocheck.$$

Commented by iloveisrael last updated on 09/Jul/21

$$\mathrm{chek}\mathrm{your}\mathrm{working}$$$$\mathrm{you}\mathrm{write}{\mathrm{v}}^2+\mathrm{v}⩾0$$$$\mathrm{and}-1⩽\mathrm{v}⩽0?\mathrm{is}\mathrm{it}\mathrm{correct}?$$$$\mathrm{hmm}..$$

Commented by mr W last updated on 09/Jul/21

$$ihavesaidyouweretofastwithyour$$$$comment.youcommentedbeforei$$$$fixedmytypos.v^2+v⩾0wastypo.itis$$$$infact{v}^2+v⩽0.ifyoucheckthoroughly,$$$$youhavehadknownthatitisatypo.$$

Commented by mr W last updated on 09/Jul/21

$$besidesishowedwiththegraphthat$$$$myansweriscorrect.$$

Commented by mathdanisur last updated on 09/Jul/21

$$alotcoolSer,thankyou$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com