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# Finding a function

## Question

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

## 4 answers to this question

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25 minutes ago, plasmid said:

Sort of combining the two solutions already given

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f(x) = C 3log2x

where C is any constant

f(2x) = C 3log2(2x) = C 31+log2x = C (3 3log2x) = 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) = 3log2(|x|)almost works, but still has a discontinuity at x = 0.

Edited by ThunderCloud

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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.

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Spoiler

f(x) = 0

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Sort of combining the two solutions already given

Spoiler

f(x) = C 3log2x

where C is any constant

f(2x) = C 3log2(2x) = C 31+log2x = C (3 3log2x) = 3f(x)

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