Finding a function

[BMAD]

Find a continuous function where the following identity is true: f(2x) = 3f(x)

[Buddyboy3000]

Spoiler

I'm not too sure if this is right, but I got two possible solutions:

y=3^(log(x)/log(2))

            and

y=-3^(log(-x)/log(2))

I got these two solutions by trying to make an equation where the y-value triples only when the x-value doubles, and not after each x-value.

 

[ThunderCloud]

You asked for it...

 

Spoiler

f(x) = 0

 

[plasmid]

Sort of combining the two solutions already given

Spoiler

f(x) = C 3^log₂x

where C is any constant

 

f(2x) = C 3^log₂(2x) = C 3^1+log₂x = C (3 3^log₂x) = 3f(x)

 

[ThunderCloud]

25 minutes ago, plasmid said:

Sort of combining the two solutions already given

  Hide contents

f(x) = C 3^log₂x

where C is any constant

 

f(2x) = C 3^log₂(2x) = C 3^1+log₂x = C (3 3^log₂x) = 3f(x)

 

Spoiler

I suspect the issue with this solution is that f(x) is not defined for all x, and therefore not continuous for all x.

f(x) = 3^log2(|x|)almost works, but still has a discontinuity at x = 0.

 

Edited December 7, 2017 by ThunderCloud