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 217541 by hardmath last updated on 15/Mar/25

$$\mathrm{f}:\mathbb{R}\to \mathbb{R}$$$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)={\mathrm{x}}^2-\mathrm{x}+1$$$$\mathrm{f}\left(0\right)=?$$

Answered by mr W last updated on 16/Mar/25

$$sayf\left(0\right)=k$$$$f\left(k\right)=f\left(f\left(0\right)\right)=0^2-0+1=1$$$$$$$$replacexwithf\left(x\right):$$$$f\left(f\left(f\left(x\right)\right)\right)=f^2\left(x\right)-f\left(x\right)+1$$$$\underset{―}{f(x^2-x+1)=f^2\left(x\right)-f\left(x\right)+1}$$$$setx=0:$$$$f\left(1\right)=f^2\left(0\right)-f\left(0\right)+1$$$$\Rightarrow f\left(1\right)=k^2-k+1$$$$$$$$setx=1:$$$$f\left(1\right)=f^2\left(1\right)-f\left(1\right)+1$$$${(f\left(1\right)-1)}^2=0$$$$\Rightarrow f\left(1\right)=1$$$$$$$$k^2-k+1=1$$$$\Rightarrow k(k-1)=0$$$$\Rightarrow k=0or1$$$$ifk=0,i.e.f\left(0\right)=0,$$$$butf\left(k\right)=f\left(0\right)=1\ne 0$$$$\Rightarrow k=0isnotvalid.$$$$sok=f\left(0\right)=1istheonlysolution.$$

Commented by hardmath last updated on 16/Mar/25

$$\mathrm{thankyou}\mathrm{dear}\mathrm{professor}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com