1. ## 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

2. 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:

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

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

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

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

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

3. 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 $a$ 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

4. 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?

5. 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.

6. 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

7. Originally Posted by Jhevon
Nice. it's a great quote