# Arbitrary powers

Printable View

• Dec 8th 2007, 01:47 PM
akhayoon
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

:confused:
• Dec 8th 2007, 02:01 PM
Jhevon
Quote:

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

:confused:

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

note that:

$f(x) = \frac 12 \left( a^x + a^{-x} \right)$

$f(y) = \frac 12 \left( a^y + a^{-y} \right)$

$f(x + y) = \frac 12 \left( a^{x + y} + a^{-x - y} \right)$

$f(x - y) = \frac 12 \left( a^{x - y} + a^{y - x} \right)$

now put those together as the equation suggests and see what you get
• Dec 8th 2007, 02:12 PM
Constatine11
Quote:

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

:confused:

This is a bit of generalised triganometry

Suppose for some $a$ that:

$f(x)=(1/2) (x^x+a^{-x})$

Then:

$
f(x+y)+f(x-y) = \frac{1}{2}\left[ a^{x+y}+a^{-x-y}\right]+\frac{1}{2}\left[ a^{-x-y}+a^{-x+y}\right]$

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

ZB
• Dec 8th 2007, 02:15 PM
Jhevon
Quote:

Originally Posted by Constatine11
Suppose for some $a$ that:

$f(x)=(1/2) (x^x+a^{-x})$

Then:

$
f(x+y)+f(x-y) = \frac{1}{2}\left[ a^{x+y}+a^{-x-y}\right]+\frac{1}{2}\left[ a^{-x-y}+a^{-x+y}\right]$

.................. =\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?
• Dec 8th 2007, 03:05 PM
ThePerfectHacker
Quote:

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.
• Dec 8th 2007, 03:13 PM
Jhevon
Quote:

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
• Dec 8th 2007, 03:20 PM
ThePerfectHacker
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.