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Question Number 3789 by 123456 last updated on 21/Dec/15

f(nx)=f(x+n)−f(x)  f(0)=0  f(x)=?

$${f}\left({nx}\right)={f}\left({x}+{n}\right)−{f}\left({x}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)=? \\ $$

Commented by Rasheed Soomro last updated on 21/Dec/15

f(nx)=f(x+n)−f(x)  −−−−  x=0  f(0)=f(n)−f(0)  f(n)=0  n  is  an  arbitrary constant  So  f(x)=0    ???

$${f}\left({nx}\right)={f}\left({x}+{n}\right)−{f}\left({x}\right) \\ $$$$−−−− \\ $$$${x}=\mathrm{0} \\ $$$${f}\left(\mathrm{0}\right)={f}\left({n}\right)−{f}\left(\mathrm{0}\right) \\ $$$${f}\left({n}\right)=\mathrm{0} \\ $$$${n}\:\:{is}\:\:{an}\:\:{arbitrary}\:{constant} \\ $$$${So} \\ $$$${f}\left({x}\right)=\mathrm{0}\:\:\:\:??? \\ $$

Commented by Rasheed Soomro last updated on 21/Dec/15

f(nx)=f(x+n)−f(x)  n=1  f(x)=f(1+n)−f(x)  f(x)=((f(1+x))/2)  x=0 in above  f(0)=((f(1+0))/2)=0  f(1)=0  f(1)=((f(2))/2)=0  f(2)=0  f(3)=0  ⋮  f(x)=0

$${f}\left({nx}\right)={f}\left({x}+{n}\right)−{f}\left({x}\right) \\ $$$${n}=\mathrm{1} \\ $$$${f}\left({x}\right)={f}\left(\mathrm{1}+{n}\right)−{f}\left({x}\right) \\ $$$${f}\left({x}\right)=\frac{{f}\left(\mathrm{1}+{x}\right)}{\mathrm{2}} \\ $$$${x}=\mathrm{0}\:{in}\:{above} \\ $$$${f}\left(\mathrm{0}\right)=\frac{{f}\left(\mathrm{1}+\mathrm{0}\right)}{\mathrm{2}}=\mathrm{0} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{1}\right)=\frac{{f}\left(\mathrm{2}\right)}{\mathrm{2}}=\mathrm{0} \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{0} \\ $$$$\vdots \\ $$$${f}\left({x}\right)=\mathrm{0} \\ $$

Commented by 123456 last updated on 21/Dec/15

f(x)=((f(x+1))/2)

$${f}\left({x}\right)=\frac{{f}\left({x}+\mathrm{1}\right)}{\mathrm{2}} \\ $$

Commented by prakash jain last updated on 21/Dec/15

n cannot be treated as variable since  it is a constant for all x.  for example  g(x)=x−n  g(n)=0 but g(x)≠0

$${n}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{treated}\:\mathrm{as}\:\mathrm{variable}\:\mathrm{since} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{for}\:\mathrm{all}\:{x}. \\ $$$${for}\:{example} \\ $$$${g}\left({x}\right)={x}−{n} \\ $$$${g}\left({n}\right)=\mathrm{0}\:{but}\:{g}\left({x}\right)\neq\mathrm{0} \\ $$

Commented by Rasheed Soomro last updated on 21/Dec/15

ThankS!

$$\mathcal{T}{hank}\mathcal{S}! \\ $$

Commented by Rasheed Soomro last updated on 21/Dec/15

for x=0  ,  f(nx)=f(x+n)−f(x)⇒f(n)=0    What is meant by  definition  f(n)=0  Couldn′t we use this definition for  n=1,n=4,n=anyvalue  Couldn′t we say  f(6)=0, f(anyvalue)=0  If  yes then  n is just like a variable?

$${for}\:{x}=\mathrm{0}\:\:,\:\:{f}\left({nx}\right)={f}\left({x}+{n}\right)−{f}\left({x}\right)\Rightarrow{f}\left({n}\right)=\mathrm{0}\:\: \\ $$$$\mathcal{W}{hat}\:{is}\:{meant}\:{by}\:\:{definition}\:\:{f}\left({n}\right)=\mathrm{0} \\ $$$${Couldn}'{t}\:{we}\:{use}\:{this}\:{definition}\:{for} \\ $$$${n}=\mathrm{1},{n}=\mathrm{4},{n}={anyvalue} \\ $$$${Couldn}'{t}\:{we}\:{say}\:\:{f}\left(\mathrm{6}\right)=\mathrm{0},\:{f}\left({anyvalue}\right)=\mathrm{0} \\ $$$${If}\:\:{yes}\:{then}\:\:{n}\:{is}\:{just}\:{like}\:{a}\:{variable}? \\ $$

Answered by Rasheed Soomro last updated on 21/Dec/15

f(nx)=f(x+n)−f(x)...................(i)  f(0)=0  f(x)=?  −−−−−−−−  n=1  in   (i)  f(x)=f(x+1)−f(x)⇒2f(x)=f(x+1)  x→x−1 in above  f(x)=2f(x−1)  f(0)=0  [given]  f(1)=2f(0)=2(0)=0  f(2)=2f(1)=0  f(3)=2f(2)=0    f(x)=0  This can be proved by mathematical induction.

$${f}\left({nx}\right)={f}\left({x}+{n}\right)−{f}\left({x}\right)...................\left({i}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)=? \\ $$$$−−−−−−−− \\ $$$${n}=\mathrm{1}\:\:{in}\:\:\:\left({i}\right) \\ $$$${f}\left({x}\right)={f}\left({x}+\mathrm{1}\right)−{f}\left({x}\right)\Rightarrow\mathrm{2}{f}\left({x}\right)={f}\left({x}+\mathrm{1}\right) \\ $$$${x}\rightarrow{x}−\mathrm{1}\:{in}\:{above} \\ $$$${f}\left({x}\right)=\mathrm{2}{f}\left({x}−\mathrm{1}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0}\:\:\left[{given}\right] \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2}{f}\left(\mathrm{0}\right)=\mathrm{2}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{2}{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{2}{f}\left(\mathrm{2}\right)=\mathrm{0} \\ $$$$ \\ $$$${f}\left({x}\right)=\mathrm{0} \\ $$$${This}\:{can}\:{be}\:{proved}\:{by}\:{mathematical}\:{induction}. \\ $$

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