Thread: Arbitrary powers

the question states

f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

Originally Posted by akhayoon

the question states

f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

did you write out the equation to see if you got any ideas? what have you tried?

note that:









now put those together as the equation suggests and see what you get

Originally Posted by akhayoon

the question states

f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

This is a bit of generalised triganometry

Suppose for some that:



Then:



..................

ZB

Originally Posted by Constatine11

Suppose for some that:



Then:



.................. =\frac{1}{2} a^x \left[ a^{y}+a^{-y}\right]+\frac{1}{2} a^x \left[ a^{y}+a^{-y}\right]

where did that quote in your signature come from?

Originally Posted by Jhevon

where did that quote in your signature come from?

The quote comes from Isaac Newton the "giants" he is referring to are the mathematicians and astronomers of the past: Kepler, Galileo, probably Archimedes, but the most important giant that inspired Newton the most was Fermat.

Originally Posted by ThePerfectHacker

The quote comes from Isaac Newton the "giants" he is referring to are the mathematicians and astronomers of the past: Kepler, Galileo, probably Archimedes, but the most important giant that inspired Newton the most was Fermat.

Nice. it's a great quote

Originally Posted by Jhevon

Nice. it's a great quote

How about:
"And perhaps posterity shall thank me for having shown the Ancients did not know everything" (Fermat).

I am not exactly sure what I means, but it sounds like a King James type of quotation which are really eqoulently stated. I wish I could talk like that (in old English) but I am not so good.