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Question Number 2504 by 123456 last updated on 21/Nov/15
f(x,y)=f(x,y−x) f(x,y)=f(y,x) f(0,y)=y^2 does g(x):=f(x,x) g(−x)=^? g(x) f(10,5)=?
$${f}\left({x},{y}\right)={f}\left({x},{y}−{x}\right) \\ $$$${f}\left({x},{y}\right)={f}\left({y},{x}\right) \\ $$$${f}\left(\mathrm{0},{y}\right)={y}^{\mathrm{2}} \\ $$$$\mathrm{does} \\ $$$${g}\left({x}\right):={f}\left({x},{x}\right) \\ $$$${g}\left(−{x}\right)\overset{?} {=}{g}\left({x}\right) \\ $$$${f}\left(\mathrm{10},\mathrm{5}\right)=? \\ $$
Answered by Rasheed Soomro last updated on 21/Nov/15
f(x,y)=f(x,y−x)...............(i) f(x,y)=f(y,x)....................(ii) f(0,y)=y^2 ...........................(iii) does g(x):=f(x,x).........................(iv) g(−x)=^? g(x) f(10,5)=? −−−−−−−−−−−−−−−− By (ii) and (i) f(10,5)=f(5,10)=f(5,10−5)=f(5,5) By (i) again f(5,5)=f(5,5−5)=f(5,0) By (ii) f(5,0)=f(0,5) By (iii) f(0^� ,5)=25 Hence f(10,5)=25 g(x)=f(x,x)=f(x,0) By (i) f(x,0)=f(0,x)=x^2 By (ii) and (iii) So g(x)=x^2 and g(−x)=(−x)^2 =x^2 =g(x) or g(−x)=g(x)
$${f}\left({x},{y}\right)={f}\left({x},{y}−{x}\right)……………\left({i}\right) \\ $$$${f}\left({x},{y}\right)={f}\left({y},{x}\right)………………..\left({ii}\right) \\ $$$${f}\left(\mathrm{0},{y}\right)={y}^{\mathrm{2}} \:\:………………………\left({iii}\right) \\ $$$$\mathrm{does} \\ $$$${g}\left({x}\right):={f}\left({x},{x}\right)…………………….\left({iv}\right) \\ $$$${g}\left(−{x}\right)\overset{?} {=}{g}\left({x}\right) \\ $$$${f}\left(\mathrm{10},\mathrm{5}\right)=? \\ $$$$−−−−−−−−−−−−−−−− \\ $$$${By}\:\left({ii}\right)\:{and}\:\left({i}\right)\:\:\:{f}\left(\mathrm{10},\mathrm{5}\right)={f}\left(\mathrm{5},\mathrm{10}\right)={f}\left(\mathrm{5},\mathrm{10}−\mathrm{5}\right)={f}\left(\mathrm{5},\mathrm{5}\right) \\ $$$${By}\:\left({i}\right)\:{again}\:{f}\left(\mathrm{5},\mathrm{5}\right)={f}\left(\mathrm{5},\mathrm{5}−\mathrm{5}\right)={f}\left(\mathrm{5},\mathrm{0}\right) \\ $$$${By}\:\left({ii}\right)\:\:{f}\left(\mathrm{5},\mathrm{0}\right)={f}\left(\mathrm{0},\mathrm{5}\right) \\ $$$${By}\:\left({iii}\right)\:{f}\left(\bar {\mathrm{0}},\mathrm{5}\right)=\mathrm{25} \\ $$$${Hence}\:{f}\left(\mathrm{10},\mathrm{5}\right)=\mathrm{25} \\ $$$${g}\left({x}\right)={f}\left({x},{x}\right)={f}\left({x},\mathrm{0}\right)\:\:{By}\:\left({i}\right) \\ $$$$\:\:\:\:\:{f}\left({x},\mathrm{0}\right)={f}\left(\mathrm{0},{x}\right)={x}^{\mathrm{2}} \:\:\:\:{By}\:\left({ii}\right)\:{and}\:\left({iii}\right) \\ $$$${So}\:{g}\left({x}\right)={x}^{\mathrm{2}} \\ $$$${and}\:{g}\left(−{x}\right)=\left(−{x}\right)^{\mathrm{2}} ={x}^{\mathrm{2}} ={g}\left({x}\right) \\ $$$${or}\:\:\:{g}\left(−{x}\right)={g}\left({x}\right) \\ $$
Commented by prakash jain last updated on 21/Nov/15
Elegant!
$$\mathrm{Elegant}! \\ $$
Commented by Rasheed Soomro last updated on 21/Nov/15
THANK^(SSsss) !
$$\mathcal{THANK}^{\mathcal{SS}{sss}} ! \\ $$

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