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 155500 by VIDDD last updated on 01/Oct/21

Answered by MJS_new last updated on 01/Oct/21

$$f(x+y)-(f\left(x\right)+f\left(y\right))=0$$$$2axy-c=0$$$$\mathrm{this}\mathrm{is}\mathrm{only}\mathrm{true}\mathrm{\forall }x,y\in \mathbb{R}\mathrm{with}a=0\Rightarrow c=0$$$$a+b+c=3\Rightarrow b=3$$$$f\left(x\right)=3x$$$$\mathrm{the}\mathrm{rest}\mathrm{is}\mathrm{easy}$$

Commented by VIDDD last updated on 01/Oct/21

$$greaysir;thanks$$

Commented by puissant last updated on 01/Oct/21

$$f\left(n\right)=3n$$$$\Rightarrow \underset{n=1}{\sum ^{10}}f\left(n\right)=\underset{n=1}{\sum ^{10}}3n=3\underset{n=1}{\sum ^{10}}n=3\bullet \frac{10\times 11}{2}$$$$=3\times 55=165...$$

Answered by Rasheed.Sindhi last updated on 02/Oct/21

$$\mathbb{A}\mathrm{n}\mathbb{A}\mathrm{lternate}\mathbb{W}\mathrm{ay}$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c},\mathrm{where}\mathrm{a}+\mathrm{b}+\mathrm{c}=3$$$$\mathrm{and}\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right).$$$$\mathrm{f}\left(0\right)=\mathrm{c}$$$$\mathrm{f}(0+0)=\mathrm{f}\left(0\right)+\mathrm{f}\left(0\right)$$$$\mathrm{c}=\mathrm{c}+\mathrm{c}\Rightarrow \mathrm{c}=0\Rightarrow \mathrm{f}\left(0\right)=0$$$$\mathrm{a}+\mathrm{b}+\mathrm{c}=3\Rightarrow \mathrm{a}+\mathrm{b}=3.......\mathrm{A}$$$$\mathrm{f}\left(1\right)=3$$$$\mathrm{f}(-1)=\mathrm{a}{(-1)}^2+\mathrm{b}(-1)+0=\mathrm{a}-\mathrm{b}$$$$\mathrm{f}(-1+1)=\mathrm{f}(-1)+\mathrm{f}\left(1\right)=\mathrm{f}(-1)+3$$$$\mathrm{f}\left(0\right)=0=\mathrm{f}(-1)+3\Rightarrow \mathrm{f}(-1)=-3$$$$\mathrm{a}-\mathrm{b}=-3..........................\mathrm{B}$$$$\mathrm{A}\mathrm{\&}\mathrm{B}\Rightarrow \mathrm{a}=0,\mathrm{b}=3$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}={0\mathrm{x}}^2+3\mathrm{x}+0$$$$\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com