Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 128424 by bramlexs22 last updated on 07/Jan/21

If f(x) is an even function and  satisfies the relation x^2 f(x)−2f(x)=g(x)  where g(x) is an odd function   then the value of f(5) is   (a) 0  (b) ((37)/(75))   (c) 4    (d) ((51)/(77))

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function}\:\mathrm{and} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{x}^{\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)−\mathrm{2f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\mathrm{where}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function}\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{5}\right)\:\mathrm{is}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{0}\:\:\left(\mathrm{b}\right)\:\frac{\mathrm{37}}{\mathrm{75}}\:\:\:\left(\mathrm{c}\right)\:\mathrm{4}\:\:\:\:\left(\mathrm{d}\right)\:\frac{\mathrm{51}}{\mathrm{77}} \\ $$

Answered by mr W last updated on 07/Jan/21

x^2 f(x)−2f(x)=g(x)   ...(i)  (−x)^2 f(−x)−2f(−x)=g(−x)  x^2 f(x)−2f(x)=−g(x)   ...(ii)  (i)+(ii):  2(x^2 f(x)−2f(x))=0  (x^2 −2)f(x)=0  (5^2 −2)f(5)=0  ⇒f(5)=0

$${x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)={g}\left({x}\right)\:\:\:...\left({i}\right) \\ $$$$\left(−{x}\right)^{\mathrm{2}} {f}\left(−{x}\right)−\mathrm{2}{f}\left(−{x}\right)={g}\left(−{x}\right) \\ $$$${x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)=−{g}\left({x}\right)\:\:\:...\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right): \\ $$$$\mathrm{2}\left({x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)\right)=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{2}\right){f}\left({x}\right)=\mathrm{0} \\ $$$$\left(\mathrm{5}^{\mathrm{2}} −\mathrm{2}\right){f}\left(\mathrm{5}\right)=\mathrm{0} \\ $$$$\Rightarrow{f}\left(\mathrm{5}\right)=\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com