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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

Let $f(x)=2^{kx}+9$ , where $k$ is a real number. If $f(3):f(6)=1:3$ , determine the value of $f(9)-f(3)$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.